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A review of multivariate orthogonal polynomials
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Review Paper
A review of multivariate orthogonal polynomialsR Mourad E.H. Ismail a, Ruiming Zhang b,∗ a b
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA College of Science, Northwest A&F University Yangling, Shaanxi 712100, PR China
a r t i c l e
i n f o
Article history: Received 24 September 2016 Accepted 23 November 2016 Available online xxx MSC: 33C50 33D50 33C45 33D45
a b s t r a c t This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user. © 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Keywords: Disc polynomials Zernike polynomials 2D-Laguerre polynomials q-2D-Laguerre polynomials Generating functions Ladder operators q-Sturm–Liouville equations q-integrals q-Zernike polynomials 2D-Jacobi polynomials q-2D-Jacobi polynomials Connection relations Biorthogonal functions Generating functions Rodrigues formulas Zeros
1. Introduction This is a survey paper of certain classes of orthogonal polynomials in two complex variables and the multivariate Hermite polynomials. The multivariate real Hermite polynomials go back to Hermite and they were studied in some detail in the book by Appell and Kampé de Feriet [4]. We felt it will be beneficial to collect scattered old and new results on 2D and nD polynomials in one place. This paper complements and updates the influential survey
R Part of this work was completed at BIRS in Banff and the authors acknowledge the support from BIRS. Research of R. Zang was partially supported by Chinese national natural science foundation, Grant no. 11371294. ∗ Corresponding author. E-mail address: [email protected] (R. Zhang).
article [40] by Koorwinder which is now 40 years old but is still a valuable reference on the subject. One of the earliest orthogonal polynomials are the Legendre polynomials which are orthogonal with respect to dx = (Lebesque measure) on the unit interval [−1, 1]. They are the spherical harmonics in R2 . The ultraspherical (or Gegenbauer) polynomials are the spherical harmonics on Rm and are orthogonal with respect ν +3/2 ) to √( (1 − x2 )ν dx on the unit interval [−1, 1]. If we replace x π (ν +1 ) √ 2 by x/ ν and let ν → ∞ the measure becomes π −1/2 e−x dx and the ultraspheical polynomials, properly scaled, tend to Hermite polynomials. The next sequence of polynomials in this hierarchy are the Jacobi polynomials which are orthogonal on the unite interval with respect to
2−α −β −1 (α +β +2 ) α (α +1 )(β +1 ) (1 − x ) (1 +
x )β dx. The Laguerre poly-
nomials arise when x is replaced by −1 + 2x/α and α → ∞. It is worth mentioning that none of the one variable polynomials is
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Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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named after the person who first discovered them, see the interesting paper by Askey [5]. The 2D history is somewhat parallel. In the late 1920s Frits Zernike (Nobel prize for physics in 1953) was working on optical problems involving telescopes and microscopes. He introduced polynomials orthogonal on |z| ≤ 1 with respect to the area measure, so this is the 2D analogue of Legendre polynomials, see [58] and [59]. The more general disc or Zernike polynomials are orthogonal with respect to (1 − x2 − y2 )ν dxdy on the unit disc. Again √ √ if we replace (x, y) by (x/ ν , y/ ν ) and let ν → ∞ this measure, 2 2 properly normalized becomes e−x −y dxdy on R2 . The polynomials become the 2D-Hermite polynomials introduced by Ito in [34] in a different way. They are defined by
Hm,n (z1 , z2 ) =
m ∧n k=0
m k
n (−1 )k k!z1m−k z2n−k , k
f (z ) − f (qz ) (Dq,z f )(z ) = , ( 1 − q )z 1 − qα [α ]q = , 1−q
(δq,z f )(z ) = z(Dq,z f )(z ),
m ∨ n = max{m, n},
Aq (z ) :=
∞ (−z )n n2 q , ( q; q )n
(1.5)
n=0
ϑ4 (z; q ) :=
∞
(−1 )n qn zn . 2
(1.6)
n=−∞
Note that our ϑ4 is slightly different from ϑ4 in Whittaker and Watson [53]. The q-Laguerre polynomials are [25, Section 21.8] (α )
Ln
n (qα+1 ; q )n n qk ( k+α ) (z ) = (−z )k . ( q; q )n k (qα +1 ; q )k q
(1.7)
k=0
2. The complex Hermite polynomials
(1.2)
m ∧ n = min{m, n} (1.3)
and
∂ , δz = z∂z , δq,z = zDq,z . ∂z
For completeness we recall the definition of the Ramanujan function Aq (z) and ϑ4 ,
(1.1)
where a∧b := min {a, b}. In 1966 Myrick [43] considered only the radial part of the polynomials orthogonal with respect to the Jacobi type measure (x2 + y2 )α (1 − x2 − y2 )β dxdy on the unit disc. An addition theorem for the disc polynomials were found by Sa˘ piro [48] and Koornwinder [39]. Maldonado [42] very briefly considered a class of radial functions which can be used to generate 2D orthogonal polynomials. Floris [15] introduced a q-analogue of the disc polynomials proved an addition theorem for them, see also Floris and Koelink [17]. Wünche [57] also studied the disc polynomials. We shall use the notations
∂z =
[m5G;January 27, 2017;14:59]
The complex Hermite polynomials are the usual Hermite polynomials {Hn (x + iy )}n≥0 , x, y ∈ R when viewed as functions of the two variables x and y. Many of the algebraic properties of Hermite polynomials {Hn (x)}n ≥ 0 hold when x is taken as a complex variable. In particular we have the generating function [25,45,50] ∞ Hn (z ) n t = exp(2zt − t 2 ). n!
(2.1)
n=0
Theorem 2.1. Assume that
1 1 =1+ . a b
0 < a < b,
(2.2)
Then the complex Hermite polynomials satisfy the orthogonality relation, [36,52],
R2
Hm (x + iy )Hn (x − iy ) e
−ax2 −by2
π
a+b dx dy = √ 2 n! ab ab n
δm,n (2.3)
(1.4)
In addition we shall follow the standard notation for shifted factorials, hypergeometric functions and their q-analogues as in [3,19], and [25]. In Section 2 we treat the Complex Hermite polynomials {Hn (x + iy )}n≥0 . They are orthogonal on R2 and are polynomials in the two real variables x and y with complex coefficients. We state their orthogonality relations and show that they are eigenfunctions of a 2D-Laplace transform. In Section 3 we treat Ito’s 2D-Hermite polynomials in some detail. This includes orthogonality, differential recurrence relations, pure recurrence relations as well as linear and multilinear generating functions. Many of the multilinear generating functions are kernels of integral operators with explicit eigenvalues and eigenfunctions. This is typical of the material treating specific sets of orthogonal polynomials in the subsequent sections. This section is followed by Section 4 which is a systematic study of their q-analogues introduced by the present authors in [33]. Section 5 contains a survey of the n-dimensional Hermite polynomials introduced by C. Hermite in 1860s. They form a biorthogonal system of polynomials. Section 5 contains recent results from [28]. In particular we mention combinatorial interpretations of the integrals of products of the polynomials times their weight functions. In Section 6 we briefly describe a class of polynomials we introduced in [31]. This contains a general construction of bivariate orthogonal polynomials from a univariate system of polynomials orthogonal on a half line. Sections 7–11 contains important examples of this construction. In each section a specific system of orthogonal polynomials is introduced and its important properties are recorded. The paper concludes with Sections 12 where we discuss polynomial solutions to certain second order partial differential equations.
n
as well as the orthogonality relations [22]
Hm (x + iy )Hn (x + iy )e−ax −by dx dy π 2 2 = Hm (x − iy )Hn (x − iy )e−ax −by dx dy= √ 2n n! δm,n . (2.4) R2 ab 2
2
R2
The proof follows from the generating function (2.1). 2 2 The weight function e−ax −by , on R has the moments
R2
R2
(x ± iy )2n+1 e−ax (x ± iy )2n e−ax
2
2
−by2
−by2
d x d y = 0,
π
d x d y = √ ( 1/2 )n , ab
(2.5)
provided that the condition (2.2) is satisfied. Ismail and Simeonov [27] studied the combinatorics of the complex Hermite polynomials. One of their combinatorial results established the more general orthogonality relation
R2
Hm (α x + β y )Hn (γ x + δ y )e−ax
π
= √ 2n n! ab
αγ a
+
βδ
2
−by2
dx dy
n
b
δm,n
(2.6)
where
a > 0,
b > 0,
α2 a
+
β2 b
=
γ2 a
+
δ2 b
= 1.
(2.7)
Formula (2.6) implies that for
√ α/ a √ δ/ a
√ β/ b √ ∈ SO(2, R ), δ/ b
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then Hn (α x + β y ) is orthogonal to Hm (δ x + γ y ) unless m = n = 0. It is also clear that the orthogonality relations (2.3) and (2.4) are special cases of (2.6). Theorem 2.2. Suppose that we have two colored multisets. The first, of color I, has size m = (m1 , m2 , . . . , mk ) ∈ Nk0 and the second, of color II, has size n = (n1 , n2 , . . . , nk ) ∈ Nk0 . We match elements from different sets, and to each pair of matched elements we assign weight 1 if the elements have the same color and weight 1/a + 1/b if the elements have different colors. Then the total weight of all perfect matchings of this type is the number B(m, n) defined by
B ( m, n ) =
√ ab
π
2 − 2 ( | m| +| n| ) 1
k R2
× e−ax
2
−by2
euz1 +vz2 =euv
m,n≥0
then,
z1j z2k =
m, n, ≥ 0, m + = j, n + = k
Hn (z ),
(2.9)
√ ab 2 2 2 2 eiz(ξ −iη )+(z +(ξ −iη ) )/2 Hn (ξ − iη ) e−aξ −bη dξ dη √ π 2 R2 = in Hn (z ),
(2.10)
n ∈ N0 , where a and b satisfy conditions (2.2). This theorem indicates that the complex Hermite polynomials are eigenfunctions of a 2-dimensional Fourier transform. However, ∞ we could not use the orthogonal system S = {Hn (x + iy )}n=1 ∪ {1} ∪ ∞ {Hn (x − iy )}n=1 to define a 2d Fourier transform. First we observe that Hn (x − iy ) is not an eigenfunction. Furthermore, the system S √ 2 2 is not complete. It has the same L2 (R2 , πab e−ax −by dxdy ) closure ∞ ∞ as {(x + iy )n }n=1 ∪ {1} ∪ {(x − iy )n }n=1 , which does not contain all ∞ n m 2 2 of {x y }n,m=0 . For example, x , y is not in the closure. Moreover, in order to do Fourier analysis we also need both 2.3 and 2.4 under the condition 2.2 to ensure the orthogonality of system S. To require they are eigenfunctions correspond eigenvalue ±i, ±1, from Theorem 2.3 we also need to have the condition 1a + 1b = 1. But the two conditions imply that a = 1, b = ∞, which is impossible.
k n
j+k−m−n !Hm,n (z1 , z2 ) 2
j ≥ m ≥ 0, k ≥ n ≥ 0
n
j m
j ≥ m ≥ 0, k ≥ n ≥ 0 ×
j!k! Hm,n (z1 , z2 ), m!n! !
=
√ ab 2 2 2 2 eiz(ξ −iη )+(z +(ξ −iη ) )/2 Hn (ξ + iη ) e−aξ −bη dξ dη √ π 2 R2 b+a i ab
∂ j+k euz1 +vz2 j ∂ z1 ∂ z2k z1 =z2 =0
(2.8)
Theorem 2.3 [27]. The following integral relations hold
=
m,n, ≥0
=
Observe that the orthogonality relation (2.3) is the special case k = 1 of Theorem 2.2.
um vn um+ vn+ Hm,n (z1 , z2 )= Hm,n (z1 , z2 ), m!n! m!n! !
z1j z2k =
j=1
dx dy.
By 3.1 we also get
which gives
[Hm j (x + iy )Hn j (x − iy )]
3
=
j ≥ m ≥ 0, k ≥ n ≥ 0
j m
k ( j − m )!Hm,n (z1 , z2 ) n
j m
k (k − n )!Hm,n (z1 , z2 ). n
The combinatorics of these polynomials have been studied in the recent work [27]. We note that Carlitz rediscovered these polynomials in the more recent work and found their generating functions [8]. It is important to think of the 2D-Hermite polynomials as polynomials in two complex variables or four real variables but their orthogonality is on the complex plane or on R2 . From the generating function it is easy to see that the polynomials satisfy the three term recurrence relations
z1 H p,q (z1 , z2 ) = qH p,q−1 (z1 , z2 ) + H p+1,q (z1 , z2 ) z2 H p,q (z1 , z2 ) = pH p−1,q (z1 , z2 ) + H p,q+1 (z1 , z2 ).
(3.3)
They also satisfy the connection relation
m + n
Hm,n (z, z¯ ) = m!n!
i 2
m n i j+k (−1 ) j+n j!k! j=0 k=0
H j+k (x )Hm+n− j−k (y ) × , ( m − j )! ( n − k )!
(3.4)
and its inverse 3. Ito’s 2D-Hermite polynomials The complex Hermite polynomials {Hm, n (z1 , z2 )} defined by (1.1) were introduced by Ito [34] in his study of complex multiple Wiener integral and were used in [1] to study Landau levels. They were applied in [51] to coherent states, and in [55,56] to quantum optics and quasi probabilities, respectively. See also [10,20,21]. They are essentially the same as the polynomials in [11, (2.6.6)]. They also appeared in combinatorics in the work of Novik et al. [44]. Their exponential generating function is ∞ m, n=0
Hm,n (z1 , z2 )
1
π
R2
Hm,n (x + iy, x − iy )H p,q (x + iy, x − iy ) e−x
j,m≥0
r! s! is (−1 )m− j Hm,r+s−m (z, z¯ ). j ! ( r − j )! ( m − j )! ( s + j − m )! (3.5)
Also note the symmetry relation
Hm,n (z, z¯ ) = Hn,m (z, z¯ ).
(3.6)
The linearization of products is given by
Hm1 ,n1 (z1 , z2 )Hm2 ,n2 (z1 , z2 )=
n1 !n2 !Hm +m − j−k,n +n − j−k (z1 ,z2 ) 1 2 1 2 j,k j!k!(m1 − j )!(n1 −k )!(m2 −k )!(n2 − j )! ,
(3.7)
um vn = euz1 +vz2 −uv , m! n!
(3.1)
2
−y2
dx dy
= m! n! δm,p δn,q .
and its inverse is, [21],
H p+m,q+n (z1 , z2 ) p! q! m! n!
and they satisfy the orthogonality relation, [24] and [20]
Hr (x )Hs (y ) =
= (3.2)
p∧n q∧m (−1 ) j+k H p− j,q−k (z1 , z2 ) Hm−k,n− j (z1 , z2 ) . j!k! ( p − j )!(q − k )! (m − k )!(n − j )!
(3.8)
j=0 k=0
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When m = n + s ≥ n the polynomials Hm,n (z1 , z2 ) are related to Laguerre polynomials via
Theorem 3.5, finally prove Theorem 3.2 by analytic continuation. It is interesting to see this can be carried out for a larger domain.
Hn+s,n (z1 , z2 ) = (−1 )n n! z1s Ln(s ) (z1 z2 ).
Corollary 3.3. For N ∈ N, let W = ρ1 eiθ1 , . . . , ρN eiθN that ρ j > 0, θ j ∈ R for j = 1, . . . , N in (3.15), H, IN K, cj and rj are the same as in Theorem 3.1 but H is not necessarily Hermitian, then
(3.9)
The symmetry relation (3.6) shows that Hn,n (z, z¯ ) is real. The Rodrigues formulas are
Hm,n (z1 , z2 ) = (−1 )m+n ez1 z2 ∂zm2 ∂zn1 e−z1 z2 , Hm,n (z1 , z2 ) = (−1 )m+n ez1 z2 ∂zn1 ∂zm2 e−z1 z2 , where ∂z := ∂∂ . We note that
∂z
1 = 2
z
∂ ∂ −i , ∂x ∂y
∂z¯
1 = 2
(3.10)
det (IN + H )−1 exp W ∗ H (IN + H )−1W k jk c j
= K Nj=1 Nk=1 −h j,k k ,...,k 1, j
∂ ∂ +i . ∂x ∂y
n m
Hm,n (z1 , z2 ) = (−1 )n ez1 z2 ∂z1 z1 e−z1 z2 .
(3.11)
(3.12)
N, j
ρ j e iθ j
(−∂z1 + z2 ) f (z1 , z2 ) (3.13)
k=0
(3.14)
The Kibble–Slepian formula is a multilinear generating function for Hermite polynomials and is due to Kibble [37] and later was proved using Fourier transforms by Slepian [49]. Foata [18] gave a nice combinatorial proof while Louck [41] used Boson operator calculus to give a new proof of this remarkable result. The norm of a matrix A = {a j,k : 1 ≤ j, k ≤ N} is the Frobenius or the Hilbert–
N 2 Schmidt norm j,k=1 |a j,k | , [23]. Ismail [26] proved the follow-
0
u . 0
v
∞ n1,2 ,n2,1 =0
u n1,2 vn2,1 Hn ,n (z1 , z1 )Hn2,1 ,n1,2 (z2 , z2 ) n 1,2 ! n 2,1 ! 1 ,2 2 ,1
= (1 − uv )−1 exp
a
u , b
H=
Det[(IN + H )−1 ] exp ZH (IN + H )−1 Z ∗ n
(h ) j,k = K 1≤ j,k≤N nj,k ! Hr1 ,c1 (z1 , z1 ), . . . , HrN ,cN (zN , zN ),
This leads to the bilinear generating function
(3.15)
where K = (n j,k : 1 ≤ j, k ≤ N ) is a general matrix with nonnegative integer entries, ck is the sum of the elements of K in column k and rj is the sum of the elements of K in row j, that is
n j,k ,
j=1
rj =
n j,k .
Theorem 3.2. Following the notation in Theorem 3.1 we let δ be a fixed real number such that 0 < δ < 31N , then (3.15) is also true for
1 ≤ j, k ≤ N
a, b ∈ R
an1,1 bn2,2 un1,2 vn2,1 Hn +n ,n +n (z1 , z1 ) n 1,1 ! n 2,2 ! n 1,2 ! n 2,1 ! 1 ,1 1 ,2 1 ,1 2 ,1 × Hn2,1 +n2,2 ,n1,2 +n2,2 (z2 , z2 )
= [(1 + a )(1 + b) − uv]−1 × exp
(3.16)
k=1
It is important to note that the multilinear kernels given by Theorem 3.1 will always be positive. The following results, namely Theorem 3.2 and Corollary 3.3, are due to Ismail and Zhang [32].
h j,k ≤ δ,
v
n j,k ≥0,1≤ j,k≤2
j,k
ck =
(3.19)
Example 2 (General2 × 2 Case). Ismail [26] considered the general 2 × 2 case
Theorem 3.1. Let Z = (z1 , z2 , . . . , zN ), and H be an N × N Hermitian matrix with H < 1, and IN is an N × N identity matrix. Then
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 . 1 − uv
The Poisson kernel (3.19) is stated in [56] without proof. Carlitz discovered the 2D-Hermite polynomials independtly of Ito’s work and established (3.19). He did not prove the orthogonality relation.
ing
N
j
In this case Theorem 3.1 gives the Poisson kernel. Indeed in the sum in (3.15) we must have n1,1 = n2,2 = 0 and we get
is from [21] and extends a one variable result due to Burchnall [7]. Another operational formula is
N
cj
Example 1 (Poisson Kernel). As an example consider the case N = 2 with
H=
r j −c j (r j −c j ) 2 L ρ ,
where {Ln(α ) (x )} are Laguerre polynomials.
n
Hm,n (z1 , z2 ) = exp(−∂z1 ∂z2 )(z1m z2n ).
This follows from Theorem 3.2 and (3.9). It must be noted that Ismail and Zeng [30] gave a purely combinatorial proof of (3.15) but as a formal power series.
The operational formula n (−1 )k Hm,n−k (z1 , z2 ) k −m = n! ∂z1 z 1 f ( z 1 , z 2 ) , k! ( n − k )!
(3.18)
Note that (3.10) is equivalent to the formulas
Hm,n (z1 , z2 ) = (−1 )m ez1 z2 ∂zm2 z2n e−z1 z2 ,
T
(3.20)
(a(1 + b) −uv )z1 z1 + uz1 z2 + vz1 z2 + (−uv + b(1 + a ))z2 z2 . (1 + a )(1 + b) − uv
An important observation is that (3.19) is essentially a generating function for Laguerre polynomials. Write the series in (3.19) as a sum over n1,2 ≥ n2,1 plus a sum over n1,2 ≤ n2,1 minus the sum over n1,2 = n2,1 . The result is ∞
n!
( n + s )! s,n=0
(3.17)
+
It is known that a multivariate analytic function f (z1 , . . . , zn ) at (0, . . . , 0 ) can be expanded in a convergent polynomial series, [6]. Then one may prove Theorem 3.2 on a domain, possibly all of ||H|| < 1, in the following steps: first prove the polynomial series is exactly the right hand side of 3.15, then prove both sides of 3.15 are analytic inside the domain by applying the estimate in
−
and H is not necessarily Hermitian.
∞
un+s vn (z1 z¯2 )n Ln(s ) (|z1 |2 )Ln(s ) (|z2 |2 )
n!
( n + s )! s,n=0 ∞
un vn+s (z¯1 z2 )n Ln(s ) (|z1 |2 )Ln(s ) (|z2 |2 )
( uv )n Ln ( |z1 |2 )Ln ( |z2 |2 )
n=0
= (1 − uv )−1 exp
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 . 1 − uv (3.21)
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The third series can be summed by the case α = 0 of Hille–Hardy formula, [25, (4.7.20)] ∞ n=0
n! r n L(α ) (x )Ln(α ) (y ) ( α + 1 )n n
= (1 − r )
−α −1
r (x + y ) exp − 1−r
0 F1
(−; α + 1; xyr (1 − r )−2 ). (3.22)
It will be of interest to give a direct proof of (3.21) using properties of Laguerre polynomials. We call a sequence of polynomials {pm, n (z1 , z2 )} a bivariate Appell sequence if its exponential generating function is of the type ∞ m,n=0
um vn pm,n (z1 , z2 ) = A(u, v ) exp(uz1 + vz2 ), m!n!
(3.23)
where A(u, v ) is analytic in u and v and we normalize our formulas by A(0, 0 ) = 1. Theorem 3.4 [26]. The only bivariate Appell sequence pn (z1 , z2 ) such that pn (z, z¯ ) is orthogonal on R2 is {Hm,n (z, z¯ )}. For properties of Appell polynomials in one variable we refer the interested reader to [45]. Theorem 3.5 [32]. For all m, n ∈ Z+ and z ∈ C we have
|Hm,n (z, z )| ≤ e|z|
2
√ m! · n!.
Theorem 3.5 is very useful in justifying interchanging sums, or sums and integrals. We now construct some integral operators. Let
K (z1 .z¯1 , z2 , z¯2 ; u, v ) =
1
π
(1 − uv )−1 exp
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 , 1 − uv (3.24)
of (3.26) can be expanded in power series in u and v when |u| < 1 and |v| < 1. We next consider linearization of products of 2D-Hermite polynomials. Note that (3.7) can be written in the form
Hm1 ,n1 (z, z¯ )Hm2 ,n2 (z, z¯ )
n1 !Hm1 +m2 −s,n1 +n2 −s (z,z¯ ) = s m ! s! (n −s )!(m −s )! 3 F2 1
(T f )(z1 , z¯1 ) =
R2
×e
K (z1 , z¯1 , x + iy, x − iy, u, v ) f (x + iy, x − iy ) −x2 −y2
dxdy,
(3.25)
2 2 (e−x −y dxdy, R2 ).
defined on L2 We know that the polynomials 2 2 {Hr (x)Hs (y)}r, s ≥ 0 are dense in L2 (e−x −y dxdy, R2 ). Hence formula (3.5) shows that {Hm,n (z, z¯ )}m,n≥0 are also dense in the same space, with z = x + iy. Therefore the polynomial system {Hm,n (z, z¯ )}m,n≥0 −x2 −y2
is complete in L2 (e dxdy, R2 ). Then the orthogonality relation (3.2) and the symmetry (3.6) imply
R2
K (z1 , z¯1 , x + iy, x − iy, u, v )Hm,n (x + iy, x − iy )e−x
2
−y2
1
It is clear that (3.27) and the orthogonality relation (3.2) give the value of the integral
k R2
Hm j ,n j (z, z¯ )e−x
2
−y2
dxdy,
(3.28)
j=1
when k = 3. This raises the question of the evaluation of a product of k polynomials. For k > 3 this integral does not have a nice closed form but has an interesting combinatorial interpretation; see [27]. Formulas (3.7) and (3.8) are inverse relations in a combinatorial sense, see Riordan [46,47]. Another result which immediately follows from the generating function is the addition formula
Hm,n (z + w, z¯ + w¯ ) m! n! √ √ √ √ p q H j,k ( 2z, 2z¯ )) H p− j,q−k ( 2w, 2w¯ )) −( p+q )/2 =2 . j! k! ( p − j )! ( q − k )! j=0 k=0
(3.29) The above formula was proved in [21] but also follows from the exponential generating functions [26]. In [26] Ismail derived analogues of ∞
Hn+k (x )
n=0
Hn (cx ) =
tn 2 = e2xt −t Hk (x − t ), n!
n/2 k=0
n!cn−2k (1 − c2 )k Hn−2k (x ), k ! ( n − 2 k )!
[25, (4.6.29)] and [25, (4.6.33)]. His results are ∞
um vn = (−1 ) j+k euz+vz¯−uv H j,k (z − v, z¯ − v¯ ), m!n!
Hm+ j,n+k (z, z¯ )
m,n=0
(3.30) and
Hm,n (cz, c¯z¯ ) =
m ∧n
Hm− j,n− j (z, z¯ )
j=0
dxdy
= um vn Hm,n (z1 , z¯1 ).
2
−s, −m1 , −n2 −1 . n1 − s + 1, m2 − s + 1 (3.27)
and consider the integral operator
5
m! n!cm− j c¯n− j (cc¯ − 1 ) j . j! (m − j )! (n − j )! (3.31)
(3.26)
This shows that {Hm,n (z1 , z¯1 )} are the eigenfunctions of T with eigenvalues um vn and the completeness of the system implies that there are no other eigenfunctions. For 0 < u, v < 1 we observe that T is self-adjoint, compact and positive definite. But the series expansion for K cannot be obtained by applying the spectral theorem directly. The expansion obtained from spectral theorem is a re-ordering of {um vn }∞ so that the obtained 1d sequence dem,n=0 creases to 0. It is not hard to see the expansion also provides an example for the generalized Mercer’s theorem, [14]. The Fourier transform in two dimensions corresponds to the special value u = v = i in (3.26). This case was stated explicitly in [55] and can be proved by imitating the classical proof for the one variable Hermite polynomial case in [54]. One cannot however simply put u = v = i in (3.26) because the Poisson kernel formula (3.19) is valid for max{|u|, |v|} < 1, since the right hand side
Notice that by setting z = 1, formula 3.31 gives yet another representation of Hm,n (c, c¯ ) in terms of cm− j c¯n− j (cc¯ − 1 ) j . He also extended
m1 +m2 +···+mr =n
provided that
r j=1
m j =m,
= Hm,n when
r
j=1
m j!
r
j=1
j=1
j=1
j=1
a jz j,
(3.32)
j=1
m
Hm j ,n j (z j , z¯j )
n j =n j=1
r
Hm j (x j )
r 1 = Hn a jx j , n!
a2j = 1, [12, (10.13.40)], to
r
r
mj
r a j
r
a j j a¯j n j m j! n j!
a¯j z¯j ,
(3.33)
j=1
|a j |2 = 1.
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The Hermite polynomials have the generating function [45, Section 106] ∞ ( c )n n t Hn (x ) = (1 − 2xt )−c 2 F0 n! n=0
c/2, (c + 1 )/2 −4t 2 (1 − 2xt )2 . −
Then N is given by
1 N = N!
(3.34) In general (3.34) holds as a formal power series since the 2 F0 diverges, unless c is a negative integer, in which case (3.34) holds as equality between functions. Rainville [45] attributes (3.34) to Brafman for general c and to Truesdell when c = 1. We now extend (3.34) to the complex Hermite polynomials and again our result will be a formal power series identity unless a or b is a negative integer when it becomes an equality. The generating function is
∞
( a )m ( b )n m n u v H
m,n (z, z¯ ) m,n=0 m! n! a, b −a −b = (1 − uz ) (1 − vz¯ ) 2 F0
uv . − (1 − uz )(1 − vz¯ )
(3.35)
We now give a moment integral representation of the Hm, n ’s. Theorem 3.6. Let z = x + iy with x, y ∈ R. The complex Hermite polynomials have the moment representation
Hm,n (iz, iz¯ ) =
im+ n
π
R2
(r + is )m (r − is )n
× exp −(r − x )2 − (s − y )2 drds.
R2
w wn exp {−ww+iz1 w+iz2 w}drds. m
(3.37) Note that Theorems 3.6 and 3.7 are the 2D analogues of
Hn (ix ) 1 = √ ( 2i )n π
e−(y−x ) yn dy, 2
(3.38)
R
e−x R2
2
−y
2
Hm,n (cz, c¯z¯ )dxdy =
R2
(cc¯ − 1 )n n!
One more integral from [26] is
if m = n,
0
e
−x2 −y2
Hm,n (ζ , ζ¯ ) dxdy =
π if m = n.
(3.39)
0 if n − m is odd, C (m, s ) if n − m = 2s,
(3.40)
(3.41)
Several authors considered sign regularity of determinants whose entries are functions. This includes Wronskians and Turánians. One classic paper in the subject is the monumental work of Karlin and Szego˝ [35]. Theorem 3.6 is the key to proving Theorem 3.8 below. Theorem 3.8 [26]. Set z = x + iy and let N be the determinant whose elements are
{(−i )m+n+2s π Hm+s,n+s (iz, iz¯ ) : 0 ≤ m, n < N}.
k=1
r j + is j − rk − isk 2
1≤ j
2 2 e−(r j −x ) −(s j −y ) dr j ds j .
(3.42)
Hence the determinant formed by (−i )m+n (−1 )s Hm+s,n+s (iz, iz¯ ) : 0 ≤ m, n ≤ N is positive for N ≥ 0. Ismail [26] gave two extensions of the integral [45, Section 109, (4)],
Pn (x ) =
n!
2 √
π
∞
0
e−t t n Hn (xt )dt . 2
(3.43)
His two extensions of the above integral evaluation are in the following theorem. Theorem 3.9. With z = x + iy we have the integral representation
4
π
∞
0
∞
e−x
2
z (z¯ )m Hm,n (cz, cz¯ )dxdy
−y2 n
0 m n
= m!n!c c¯ 2 F1
−m, −n cc¯ − 1 , 1 cc¯
(3.44)
1
e−x
2
=
z¯ Hm2 ,n2 (z, z¯ )dxdy
−y2 m1 n1
z
R2
0
if m1 + m2 = n1 + n2 , or n2 > m1
m1 ! n1 !! ( m 1 − n 2 )!
if m1 + m2 = n1 + n2 .
(3.45)
Theorem 3.9 suggests considering the more general integrals
I ( m0 , m1 , . . . , mk ; n0 , n1 , . . . , nk ) 2 2 := π1 R2 e−x −y zm0 z¯n0 sj=1 Hm j ,n j (z, z¯ )dxdy,
z := x + iy.
Of course this integral does not have a closed form but has a combinatorial interpretation. Consider k + 1 two groups of disjoint sets S0 , S1 , . . . , Sk , T0 , T1 , . . . , Tk . The S sets are of a certain type, say type 1, and the T sets are of type 2. A perfect matching is a one to one mapping from ∪kj=0 S j onto ∪kj=0 T j . Further we allow the elements of S0 to be matched with elements of any T set, however elements of Sj , j = 0 are matched only with elements of Tr for r = j. Of course each element (vertex) is matched with a unique object. Theorem 3.10. The number of the above mentioned matchings is the integral in (3.46). 4. 2D − q-Hermite polynomials
where
ζ = ax + iby, a, b ∈ R, m = n + 2s, 2−2s π −m/2, (1 − m )/2 a2 − b2 C (m, s ) = . 2 F1 4 s+1 m! s!
N
|rk + isk |
(3.46)
see [25, (4.6.41)]. Another integral is
R2N
2s
j=1
π
Theorem 3.7. Let w = r + is with r, s ∈ R and z1 , z2 ∈ C, then we have the moment integral representation
1 π im+ n
N
and the integral evaluation
(3.36)
A two complex variable case is in [32].
e−z1 z2 Hm,n (z1 , z2 )=
×
Ismail and Zhang [33] introduced two q-analogues of the Ito polynomials. The first q-analogue is defined by
Hm,n (z1 , z2 |q ) :=
m ∧n
m k
k=0
It is clear that
q
n k
(−1 )k q(2 ) (q; q )k z1m−k z2n−k . k
(4.1)
q
Hm,n z2 , z1 q = Hn,m z1 , z2 q .
(4.2)
The second q-analogue is defined by the explicit representation
hm,n (z1 , z2 |q ) :=
m ∧n j=0
m j
q
n j
q(m− j )(n− j ) (−1 ) j (q; q ) j z1m− j z2n− j . q
(4.3)
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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7
As a basic hypergeomoetric function (4.3) takes the form
Moreover they have the Rodrigues type formulas
hm,n (z1 , z2 |q )
Hm,n (z1 , z2 |q ) =
= (−1 )n qm−n+1 ; q z1m−n 1 φ1 n
q−n q
m−n+1
q; −qm+1 z1 z2 .
(1 − 1/q )m+n qmn m Dq−1 ,z2 Dnq−1 ,z1 ( (qz1 z2 ; q )∞ ), (qz1 z2 ; q )∞
(4.4)
Note the (m, n ) − (z1 , z2 ) symmetry
(4.19) n hm,n (z1 , z2 |q ) = (q − 1 )m+n (−z1 z2 ; q )∞ Dm q,z2 Dq,z1
hm,n (z1 , z2 |q ) = hn,m (z2 , z1 |q ),
(4.5)
and the fact
.
(4.20) Furthermore we also have the operational formula
hm,n (z1 , z2 |1/q ) = q−mn i−m−n Hm,n (iz1 , iz2 |q )
(4.6)
Hm,n (z1 , z2 |q ) = ((1 − q )2 Dq,z1 Dq,z2 ; q )∞ z1m z2n , qmn
Theorem 4.1. The polynomials {Hm, n (z1 , z2 |q)} and {hm, n (z1 , z2 |q)} have the generating functions
hm,n (z1 , z2 |q ) =
∞ ( uv; q )∞ um vn = Hm,n (z1 , z2 |q ) , (uz1 , vz2 ; q )∞ m,n=0 ( q; q )m ( q; q )n
and the lowering relations
1
(−z1 z2 ; q )∞
−q1/2 uz1 , −q1/2 vz2 ; q
(−uv; q )∞
∞
(4.7)
(
−q−1
(1 − q )
2D
)
q−1 ,z1 Dq−1 ,z2 ; q ∞
(4.21)
z1m z2n .
1 − qm Hm−1,n (z1 , z2 |q ), 1−q 1 − qn Dq,z2 Hm,n (z1 , z2 |q ) = Hm,n−1 (z1 , z2 |q ), 1−q Dq,z1 Hm,n (z1 , z2 |q ) =
∞ hm,n (z1 , z2 |q ) (m−n )2 /2 m n = q u v , (q; q )m (q; q )n m,n=0
(4.8) and satisfy the functional relations
Hm,n (qz1 , z2 |q ) = Hm,n (z1 , z2 |q ) − z1 (1 − qm )Hm−1,n (z1 , z2 |q ), (4.9)
(4.22)
and
qn−m+1 (1 − qm ) hm−1,n (z1 , z2 |q ), 1−q
Dq−1 ,z1 hm,n (z1 , z2 |q ) =
(4.23)
qm−n+1 (1 − qn ) Dq−1 ,z2 hm,n (z1 , z2 |q ) = hm,n−1 (z1 , z2 |q ). 1−q
We note that (4.11) and (4.12) imply the symmetry relation
Hm,n (z1 , qz2 |q ) = Hm,n (z1 , z2 |q ) − z2 (1 − qn )Hm,n−1 (z1 , z2 |q ), (4.10) Hm,n (qz1 , z2 |q )q
−m
Hm,n (qz1 , z2 |q )q−m = Hm,n (z1 , qz2 |q )q−n .
Theorem 4.2. The 2D − q-Hermite polynomials have the raising relations
Hm+1,n (z1 , z2 |q ) = qn
= Hm,n (z1 , z2 |q ) − q−1 (1 − qm )(1 − qn )Hm−1,n−1 (z1 , z2 |q ), (4.11)
Hm,n+1 (z1 , z2 |q ) = qm
Hm,n (z1 , qz2 |q )q−n (4.12)
hm,n z1 q−1 , z2 |q =hm,n (z1 , z2 |q ) + z1 (1−qm )q−m hm−1,n (z1 , z2 |q ), (4.13) hm,n z1 , z2 q−1 |q = hm,n (z1 , z2 |q ) + z2 (1 − qn )q−n hm,n−1 (z1 , z2 |q ), (4.14) q hm,n (z1 /q, z2 |q ) m
= hm,n (z1 , z2 |q ) + (1 − q )(1 − q )q m
n
1−m−n
1 − 1/q
(qz1 z2 ; q )∞ 1 − 1/q
Dq−1 ,z2 ( (qz1 z2 ; q )∞ Hm,n (z1 , z2 |q ) ),
(qz1 z2 ; q )∞
Dq−1 ,z1 ( (qz1 z2 ; q )∞ Hm,n (z1 , z2 |q ) ). (4.25)
= Hm,n (z1 , z2 |q ) − q−1 (1 − qm )(1 − qn )Hm−1,n−1 (z1 , z2 |q ).
(4.24)
hm−1,n−1 (z1 , z2 |q ), (4.15)
q hm,n (z1 , z2 /q|q )
and
(4.16) They also satisfy the three term recurrence relations
z1 Hm,n (z1 , z2 |q ) = qm (1 − qn )Hm,n−1 (z1 , z2 |q ) + Hm+1,n (z1 , z2 |q ), z2 Hm,n (z1 , z2 |q ) = qn (1 − qm )Hm−1,n (z1 , z2 |q ) + Hm,n+1 (z1 , z2 |q ). (4.17) and
qn z1 hm,n (z1 , z2 |q ) = hm+1,n (z1 , z2 |q ) + (1 − qn )hm,n−1 (z1 , z2 |q ), qm z2 hm,n (z1 , z2 |q ) = hm,n+1 (z1 , z2 |q ) + (1 − qm )hm−1,n (z1 , z2 |q ). (4.18)
hm,n (z1 , z2 |q ) hm+1,n (z1 , z2 |q ) = (q − 1 )(−z1 z2 ; q )∞ Dq,z2 , (−z1 z2 ; q )∞ (4.26) hm,n (z1 , z2 |q ) hm,n+1 (z1 , z2 |q ) = (q − 1 )(−z1 z2 ; q )∞ Dq,z1 . (−z1 z2 ; q )∞ Moreover the same polynomials have the multiplication formulas
Hm,n (az1 , bz2 |q ) =
m ∧n j=0
m j
q
n j
Hm− j,n− j (z1 , z2 |q ) (q, 1/ab; q ) j (q; q ) j , a j−m b j−n q
(4.27)
(q, ab; q ) j am− j bn− j hm− j,n− j (z1 , z2 ; q ).
(4.28)
hm,n (az1 , bz2 ; q )
n
= hm,n (z1 , z2 |q ) + (1 − qm )(1 − qn )q1−m−n hm−1,n−1 (z1 , z2 |q ),
=
m ∧n j=0
m j
q
n j
q
This theorem can be proved using the generating functions of Theorem 4.1. The polynomials Hm, n have the representation
Hm,n z1 , z2 q = (−1 )n
n (q; q )m q(2 ) m−n z1 pn z1 z2 , qm−n q , (q; q )m−n
(4.29)
where pn (x; qα |q) is the little q-Laguerre or Wall’s polynomials, [38], defined by
pn ( x; a|q ) =
2
φ1
q−n , 0 aq
q; qx .
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Theorem 4.3. The polynomials orthogonality
Hm,n z, z q
satisfy the following
where ∞
dμ(z, z ) =
k=0
qk dθ δ (r − qk/2 ) , q ; q 2 π ( )k
dμ(z, z¯ ) ( u1 u2 v1 v2 ; q )∞ = . (u1 z, v1 z¯, v2 z, u2 z¯; q )∞ (q, u1 u2 , v1 v2 , u1 v1 , u2 v2 ; q )∞
The orthogonality relation of the h’s is given in the following theorem. Theorem 4.4. The polynomials {hm,n (z, z|q )} satisfy the orthogonality relation
hm,n (z, z|q )hs,t (z, z|q )
(−q1/2 u1 z, −q1/2 v1 z¯, −q1/2 v2 z, −q1/2 u2 z¯; q )∞ dxdy (−zz¯; q )∞ R2 (−u1 v1 , −u2 v2 , −u1 u2 , −v1 v2 ; q )∞ = π ln q−1 . ( u1 u2 v1 v2 ; q )∞
(4.33)
Theorem 4.5. The polynomials {Hm, n (z1 , z2 )} have the generating function
um (a/u; q )n vn (b/v; q )n Hm,n z1 , z2 q ( q; q )m ( q; q )n m,n=0 a/u, b/v (az1 .bz2 ; q )∞ = φ , q ; u v . (uz vz ; q ) 2 2
k
(4.36)
q
z1 , z2 q = z1m (1/z1 z2 ; q )m ,
(4.37)
(4.38)
The convergence in (4.36)–(4.38) is uniform on compact subsets C × C. The next two theorems give the Plancherel–Rotach asymptotics of the polynomials {Hm, n (z1 , z2 |q)} and {hm, n (z1 , z2 |q)}. Theorem 4.7. Let a, b, c, d ≥ 0, τ = τ (m, n; a, b, c, d ) = (a + c )m + (b + d )n and χ = χ (m, n; a, b, c, d ) = {(a + c )m + (b + d )n}, for 0 < τ (m, n; a, b, c, d) < m∧n, then
lim
m,n→∞
1 1 (q; q )∞ Hm,n z1 qam+bn− 4 , z2 qcm+dn− 4 |q (−z1 z2 )τ z1m z2n qam2 +(b+c )mn+dn2 −(m+n )/4−τ 2 /2−τ χ
= θ4 z 1 z 2 q χ ; q 1 / 2 , holds uniformly on compact subsets of the z1 and z2 planes, where ϑ4 is defined in (1.6).
az1 , bz2
hm,n w1 q−am−bn , w2 q−(1−a )m−(1−b)n |q (q; q )2∞ wm wn2 q(a−b) 1
mn−am2 +
(b−1 )
n2
1 w1 w2
.
Since the polynomial Hm,n (z, z¯ ) factors as a function of θ times a radial function it is clear that with z1 = z, z2 = z¯ the zeros of the polynomials investigated here as functions of z lie on circles. Now (4.38) implies the following asymptotic result. Theorem 4.9. Assume that the zeros of Hm,n (z, z¯|q ) lie on the circles with radii
r1 (H, m, n ) > r2 (H, m, n ) > · · · .
(4.40)
Then m,n→∞
(4.34)
= Aq
(4.39)
lim r j (H, m, n ) = q j/2 , j = 1, 2, . . . .
∞
lim z−n Hm,n n→∞ 2
m,n→∞
As it was the case with the Hm, n ’s the generating function (4.8) show that the orthogonality relation (4.32) is equivalent to the q-beta type integral
2
q(2 ) (−z1 z2 )−k = z2n (1/z1 z2 ; q )n ,
Similarly we establish the limiting relations
lim
where x = x + iy, x, y ∈ R, m, n, s, t ∈ N0 .
1
k=0
(4.32)
q
∞
n n k
Theorem 4.8. For a, b ∈ C and 0 < < 1 the following asymptotic result holds uniformly when w1 , w2 are in compact subsets of the complex plane
dxdy
(−zz; q )∞ π log q−1 (q; q )m (q; q )n = δm,s δn,t , (m−n )2 /2+(m+n )/2
R2
= z2n
m,n→∞
(4.31)
z1 , z2 q
It is clear that the orthogonality relation (4.30) and the generating function (4.7) imply the q-beta integral evaluation
C
lim z1−m z2−n Hm,n z1 , z2 q = (1/z1 z2 ; q )∞ .
and z = reiθ , r ∈ R+ , θ ∈ [0, 2π ], m, n, s, t ∈ N0 .
lim z−m Hm,n m→∞ 1
qmn (q; q )m (q; q )n Hm,n z, z q Hs,t z, z q dμ(z, z ) = δm,s δn,t , (q; q )∞ C (4.30)
[m5G;January 27, 2017;14:59]
(4.41)
Note that the limiting distribution of the radii of the circles of zeros coincides with the location of the masses of the orthogonality measure of the polynomials. Theorem 4.7 implies the following results about zeros of Hm, n .
As we saw in Sections 3 the Ito polynomials are orthogonal with respect to two different measures. The next theorem show that our Hm, n (z1 , z2 |q) share this property.
Theorem 4.10. The zeros of Hm,n z1 q(am+bn−1/4 ) , z2 q(cm+dn−1/4 ) |q
Theorem 4.6. We have the orthogonality relation
The function Aq (z) has infinity many zeros and they are all positive and simple. We denote them by
p s π j=0 k=0
×
0
(q, e2iθ , e−2iθ ; q )∞
0 < i1 ( q ) < i2 ( q ) < · · · < in ( q ) < · · · .
q−s q, q q1−p r 2
δs,p .
r1 (h, m, n ) > r2 (h, m, n ) > · · · . (4.35)
z, z q are straightforward.
(4.42)
Theorem 4.11. Assume that the zeros of hm,n (z, z¯|q ) lie on the circles with radii
H j,k (reiθ , re−iθ |q )Hs−k,p− j (reiθ , re−iθ |q ) dθ (q; q ) j (q; q )k (q; q )s−k (q; q ) p− j π r 2 p (1/r 2 ; q ) p = φ ( q; q ) p 1 1
asymptotically lie on the the curves z1 z2 qχ = q±(n+1/2 ) , n = 0, 1, 2, . . . . These curves become circles when z1 = z, z2 = z¯.
The large degree asymptotics of Hm,n Indeed (4.1), Tannery’s theorem, and the q binomial theorem show that
Then
lim q(m+n )/2 r j (h, m, n ) = 1/ i j (q ), j = 1, 2, . . . ,
m,n→∞
(4.43)
(4.44)
where {ij (q)} are the zeros of the Ramanujan function ordered as in (4.42).
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We now identify the q-Sturm–Liouville problems whose eigenfunctions are either {Hm, n (z1 , z2 |q)} or {hm, n (z1 , z2 |q)}. Using the first equations in (4.22) and (4.25) we find that Hm, n satisfy the eigenvalue equation
qz1 (1 − z1 z2 )Dq−1 ,z2 Dq,z1 f (z1 , z2 ) − Dq,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.45)
=
9
1 s 1 t ∞ hs,t (z1 , z2 |q ) q 2 emk q 2 e−mk (q; q )s (q; q )t
(4.55)
s,t=0
and when |z1 z3 | < 1, |z2 /z3 | < 1, we have
(z1 z2 ; q )∞
=
(−z1 z2 , z1 z3 , z2 /z3 ; q )∞
∞ hm,n (z1 , z2 |q ) (m+n )/2 (m−n )/2 q z3 . (q; q )m (q; q )n
m,n=0
(4.56)
with
λ = λm,n
1 − qm = −q1−n . ( 1 − q )2
(4.46)
5. The Hermite polynomials on Rn
Similarly we show that Hm,n solves the eigenvalue equation
We follow the notation in [12, Section 12.8]. First we define
qz2 (1 − z1 z2 )Dq−1 ,z2 Dq,z1 f (z1 , z2 ) − Dq,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.47)
φ (x ) = (Cx, x ),
with
λ = λm,n = −q1−m
1 − qn . ( 1 − q )2
(4.48)
Combine (4.23) and (4.26) to see that hm, n (z1 , z2 |q) satisfies the equation
(1 + z1 z2 )Dq,z2 Dq−1 ,z1 f (z1 , z2 ) −
z1 D −1 f (z1 , z2 )=λ f (z1 , z2 ), 1 − q q ,z1 (4.49)
where
λ = λm,n
1−q = qn−m+1 . ( 1 − q )2 m
(4.50)
z (1 + z1 z2 )Dq,z1 Dq−1 ,z2 f (z1 , z2 ) − 2 Dq−1 ,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.51) with
λ = λm,n =
1 − qn m−n+1 q . ( 1 − q )2
(4.52)
We note the relation m−n (m−n ) Ln n z1
hm,n (z1 , z2 |q ) = (−1 ) (q; q ) n
Rn
e−φ (x )/2+(a,x ) dx = κn eψ (a )/2 ,
j=1
m∈Nn0
n u j j=1
= e(Cx,u )−φ (u )/2 = eφ (x )/2−φ (x−u )/2 ,
Im,k := κn−1
δ (x − cqy ), y ∈ Z, c, α + 1 > 0,
m j!
= e(x,u )−ψ (u )/2 = eφ (x )/2−φ (x−C
n j=1
∂ m −φ ( x ) / 2 e , ∂x
m j and ∂∂ x := m
iθ
d σ re , re
−iθ
1 = d θ d μ ( r 2 ), 2
r ∈ R , θ ∈ [0, 2π ]
=
∞
1 s 1 t ∞ Hs,t (z1 , z2 |q ) (s−t )2 q 2 q 2 e2imk q 2 e−2imk , (q; q )s (q; q )t
Rn
−z1 z2 ; z1 e2mk , z2 e−2mk ; q
n j=1
m
∂ j . If m ∈ Z \ N , we set m 0 ∂x j j
n
m j !.
(5.6)
j=1
r Rn
Hmi (x )
i=1
s
Gki (x )e−φ (x )/2 dx,
i=1
(5.7)
{mi }ri=1
where ⊂ N0 and generating function
F (U, V ) := (4.54)
×
∞
{ki }si=1
{mi }ri=1 ⊂N0 {ki }si=1 ⊂N0
s,t=0
(z1 z2 ; q )∞
(5.4)
(5.5)
where r ∈
2
−z1 qe2imk , −z2 qe−2imk ; q
.
m ∈ Nn0 ,
Hm (x )Gk (x )e−φ (x )/2 dx = δm,k
I ({mi }ri=1 , {ki }si=1 ) := κn−1
Theorem 4.12. Assume that q = e−2k and |q| < 1, then we have
u )/2
More generally, Ismail and Simeonov [28] considered the integrals
+
is also an orthogonal measure for hm,n reiθ , re−iθ |q R+ , θ ∈ [0, 2π ].
−1
Hm (x ) = Gm (x ) = 0. The following biorthogonality relation holds [12, Section 12.8]:
etc. it is clear that our proof shows that
(5.3)
The second equality in (5.3) follows from the identity φ (x ) − 2(x, Cu ) + φ (u ) = φ (x − u ). The second equality in (5.4) is obtained by replacing u by C −1 u in (5.3). The generating function (5.3) leads to the Rodrigues formula [12, Section 12.8 (20)]
dμ(x ) = x−α wQL (x; α , c, λ )dx, x
m j! mj
Gm ( x )
where |m| =
(−x; q )∞
n u j
m∈Nn0
where Ln(α ) (z; q ) is a q-Laguerre polynomial, see (1.7). Since the moment problem associated with Ln(α ) (x; q ) is indeterminate [25, Section 21.8], they have infinitely many orthogonal measures. Let xα dμ(x) be such a measure, for example,
d μ (x ) =
(5.2)
mj
Hm (x )
(4.53)
α , λ, c > 0,
κn := (2π )n/2 −1/2
where at = (a1 , . . . , an ) ∈ Rn , dx = dx1 , . . . , dxn , and is the determinant of C, follows from diagonalizing φ and a change of variable on the normal distribution. Let m = (m1 , . . . , mn ) ∈ Nn0 . The polynomials {Hm (x)} and {Gm (x)} are defined through the generating functions [12, §12.8]
Hm (x ) = (−1 )|m| eφ (x )/2
(z1 z2 ; q ),
(5.1)
where C is a positive definite n × n matrix, xt = (x1 , . . . , xn ) ∈ Rn ,
and (x, y ) = nj=1 x j y j . The integral evaluation [12, §12.8]
Similarly we find that
ψ (x ) = (C −1 x, x ) = φ (C −1 x ),
s n i=1 j=1
i, j vm i, j
mi, j !
,
⊂ N0 . If F(U, V) is the exponential m
I ({
}
mi ri=1 ,
{ }
ki si=1
)
r n u i, j i, j i=1 j=1
mi, j ! (5.8)
where U = [ui, j ] is a r × n matrix and V = [vi, j ] is an s × n matrix.
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A formula of Feldheim [13] is
i−|m| Hm (ix ) = κn−1
n
Rn
(Cy )
mj j
e−φ (x−y )/2 dy,
m ∈ Nn0 .
j=1
(5.9) Burchnal [7] proved that
d −
dx
+ 2x
m
f (x ) =
m
m Hm−k (x ) f (k ) (x ). k
(−1 )k
k=0
(5.10)
Using a technique similar to Louck’s proof of the Kibble–Slepian formula, [41], Ismail and SImeonov also proved the following generating function. Theorem 5.2. Let S and C be positive definite n × n matrices such that I + 12 C 1/2 SC 1/2 is invertible, and let {Hm (x)} be the Hermite polynomials associated with C. Then
−1/2
(s /2 )m j,l j,l −tr(M )
× exp (1/4 ) C SC =
2
1/2
M∈Mn (N0 ) M =Mt
I +C
1/2
SC
1/2
/2
−1
x, x
Hm (x ),
m j,l !
1≤ j≤l≤n
C
1/2
∞
mj
m j!
[ψ (a )/2]m/2 Hm m!
Gm1 ,m2 ,··· ,mn (x1 , . . . , xn )
( a, x ) . [2ψ (a )]1/2
(5.16)
φm (r; α )φn (r; α )rα dμ(r, β ) = ζn (α )δm,n , α ≥ 0.
0
n
det I + C 1/2 SC 1/2 /2
(5.15)
We introduced the model of this section in [31]. Let {φ n (r; α )} be a system of orthogonal polynomials satisfying the orthogonality relation
−
1
(Ca, x ) , [2φ (a )]1/2
6. Complex 2D-systems
Theorem 5.1.
m j ∂ + (Cx ) j f (x ) ∂xj j=1 m1 mn n mj ∂k = ··· (−1 )|k| Hm−k (x ) f ( x ). ∂x kj j=1 k =0 k =0
m a j
m1 +m2 +···+mn =m j=1
=
Ismail and Simeonov [28] extended proved the following multivariate version of Burchnal’s formula. n
[φ (a )/2]m/2 Hm m!
=
It is assumed the measure μ does not depend on α but may depend on another parameter β . Let
φn (r; α ) =
n
c j (n, α )r n− j ,
c j (n, α ) ∈ R,
(6.2)
j=0
and define bivariate polynomials fm,n by
fm,n (z1 , z2 ; β ) =
z1m−n φn (z1 z2 ; m − n + β ), m ≥ n, fn,m (z2 , z1 ; β ), m < n.
∂ f (z, z¯; β ) = (m − n ) fm,n (z, z¯; β ), m ≥ n. ∂θ m,n
Therefore the functional relations
(5.11)
(δz1 − δz2 ) fm,n (z1 , z2 ; β ) = (m − n ) fm,n (z1 , z2 ; β ),
m ≥ n, (6.4)
1 − qm−n fm,n (z1 , z2 ; β ), δq,z1 − qm−n δq,z2 fm,n (z1 , z2 ; β ) = 1−q
m ≥ n,
(6.5)
Moreover Ismail and Simeonov [28] proved that Proposition 5.3. Let S = {S j }nj=1 be a multiset of n sets, where each set Sj contains m j = |S j | distinct objects, j = 1, . . . , n. Let M(m ) be the set of all inhomogeneous perfect matchings of this multiset. Then
Hm (x ) =
(−cedge )
μ∈M(m ) edge∈E (μ )
fix∈Fix(μ )
(Cx )i(fix) ,
(5.12)
where E(μ) denotes the set of all edges and Fix(μ) denotes the set of all fixed points (unmatched elements) in a matching μ ∈ M(m ), cedge = c j1 , j2 , if edge ∈ E(μ) connects elements from sets S j1 and S j2 , and i(fix ) = j if fix ∈ Sj .
With the notation m = nj=1 m j , the polynomials {Hm (x) and {Gm (x) have the Rodrigues type representations
Hm (x ) = (−1 )m exp(φ (x/2 ))
∂ exp(−φ (x/2 ), n 1 ∂ xm , . . . , ∂ xm n 1 m
(5.13) Gm (x ) = (−1 )m exp(ψ (x/2 ))
∂
∂m 1 xm ,..., 1
n ∂ xm n
exp(−ψ (x/2 ). (5.14)
One curious property of the Hermite polynomials is the following
mj
m a j
m1 +m2 +···+mn =m j=1
m j!
Hm1 ,m2 ,...,mn (x1 , . . . , xn )
(6.3)
From (6.3) it is clear that
−i
where Mn (N0 ) is the set of all n × n matrices with entries from
N0 , M = [m j,l ]nj,l=1 , tr(M ) = nj=1 m j, j is the trace of M, and m =
(m1 , . . . , mn ) with m j = m j, j + nl=1 m j,l , j = 1, . . . , n.
(6.1)
hold. Theorem 6.1 [31]. The polynomials {fm, n (z1 , z2 ; β )} satisfy the orthogonality relation
dθ d μ (r 2 ; β ) 2π = ζm∧n (|m − n| + β )δm,s δn,t ,
R2
fm,n (z, z¯; β ) fs,t (z, z¯; β )
(6.6)
for all nonnegative integers m, n, s, t, where dμ(r; β ) = r β dμ(r ). The proof follows from (6.1). Unlike general 2D orthogonal polynomials [11] the fm, n ’s satisfy three term recurrence relations. Indeed we have
z2 fm+1,n (z1 , z2 ; β ) =
c0 (n, m − n + 1 + β ) fm+1,n+1 (z1 , z2 ; β ) c0 ( n + 1 , m − n + β )
c0 (n, m − n + 1 + β )cn+1 (n + 1, m − n + β ) fm,n (z1 , z2 ; β ), c0 (n + 1, m − n + β )cn (n, m − n + β ) m ≥ n, (6.7) −
and
z1 fm,n (z1 , z2 ; β ) −
c0 (n, m − n + β ) fm+1,n (z1 , z2 ; β ) c0 (n, m + 1 − n + β )
= um,n fm,n−1 (z1 , z2 ; β ),
(6.8)
with
um,n =
c0 (n, α + 1 )c1 (n, α ) − c0 (n, α )c1 (n, 1 + α ) c0 (n − 1, 1 + α )c0 (n, 1 + α )
(6.9)
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and α = m − n + β . Assume that the three term recurrence relation satisfied by the φ n ’s is
r φn (r ; α )=an (α )φn+1 (r; α ) + cn (α )φn (r; α ) + bn (α )φn−1 (r; α ) (6.10) By comparing like powers of r we find that
c0 (n, α ) an ( α ) = , c0 ( n + 1 , α )
c1 ( n + 1 , α ) c1 (n, α ) cn ( α ) = − , c0 (n, α ) c0 ( n + 1 , α ) (6.11)
11
where
δq,r f = rDq,r f.
(6.21)
Then fm, n (z1 , z2 ), for m ≥ n and α = m − n + β , satisfies the q-partial differential equations 2 An (z1 z2 , α )δq,z f + Bn (z1 z2 , α )δq,z2 f + Cn (z1 z2 , α ) f = 0, 2
(6.22)
2 An (z1 z2 , α )δq,z f + {qα Bn (z1 z2 , α ) − 2[α ]q An (z1 z2 , α )}δq,z1 f 1
+ [α ]2q An (z1 z2 , α ) − [α ]q qα Bn (z1 z2 , α ) + q2α Cn (z1 z2 , α ) f = 0. bn ( α ) =
c0 (n, α )c2 (n, α ) − c1 (n, α )2 c0 (n − 1, α )c0 (n, α ) c0 (n, α )c2 (n + 1, α ) − c1 (n, α )c1 (n + 1, α ) − c0 ( n − 1 , α )c0 ( n + 1 , α )
(6.12)
Theorem 6.5. Assume that
wα +β ( x ) = xα wβ ( x )
The following recurrence relation follows from (6.2) and (6.3).
(6.13)
We note that the definition of the polynomials fm, n indicates that for m ≥ n it may have a trivial zero at z1 = 0 but will vanish on the curves z1 z2 = r where r is a zero of φn (r; m − n + β ). The zeros of φn (r; m − n + β ) are positive, real and simple when m ≥ n. The relationships (3.9), (4.29), and (4.53) show that the polynomials Hm, n and their q-analogues are examples of fm, n polynomials. Remark 6.2. In view of the defining Eq. (6.3) the question of finding the large m, n asymptotics of the two variate polynomials fm, n is equivalent to finding the large m, n behavior of φn (r; m − n + β ). It will be of interest to carry out this program at least for some special systems, including the Freud type polynomials orthogonal with respect to xα exp(−p(x )), where p is a polynomial with positive leading term. Theorem 6.3. Let φ n (x; α ), an (α ), bn (α ) as in (6.11) and (6.12), for m ≥ n − 1 we have
z1 fm,n (z1 , z2 ) − vm,n fm+1,n (z1 , z2 ) = um,n fm,n−1 (z1 , z2 ),
(6.14)
(6.24)
and
φn ( x ; α ) =
(z1 z2 − cn (m − n + β )) fm,n (z1 , z2 ; β ) = an (m − n + β ) fm+1,n+1 (z1 , z2 ; β ) + bn (m − n + β ) fm−1,n−1 (z1 , z2 ; β ).
1 ∂ n (wα (x )xn ), wα ( x ) x
vm,n = an (m − n ).
(6.15)
We next consider differential or q-difference equations satisfied by fm, n (z1 , z2 ). Theorem 6.4. Assume that φ n (r; α ) satisfies the second order differential equation
An (r, α )δr2 f + Bn (r, α )δr f + Cn (r, α ) f = 0,
(6.16)
fm,n (z1 , z2 ; β ) =
z m−n ∂ n w (z1 z2 )zm β z2 2 1 , z2 wβ ( z1 z2 ) ∂z2
fm+1,n+1 (z1 , z2 ; β ) =
m+1−n z2 z1
(6.26)
wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
z2 m−n z1
wβ ( z1 z2 )
,
(6.27)
∂zn1 wβ (z1 z2 )z1m fm,n (z1 , z2 ; β ) = , wβ ( z1 z2 ) ∂z1 wβ (z1 z2 ) fm+1,n (z1 , z2 ; β ) fm+1,n+1 (z1 , z2 ; β ) = wβ ( z1 z2 ) and
∂z2
m+1−n z 2
z1
(6.28)
(6.29)
wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
(6.30)
We next state a q-analogue of the previous result. Theorem 6.6. Assume that
wα +β ( x ) = wα +β ( x|q ) = xα wβ ( x|q )
(6.31)
and
φn (x; α ) = φn (x; α|q ) =
1 Dn (wα (x|q )xn ), wα (x ) q,x
n = 0, 1, . . . . (6.32)
∂ δr f = r f. ∂r
(6.17)
Then, fm, n (z1 , z2 ; β ), for α = m − n + β and m ≥ n, satisfies the second order partial differential equations
An (z1 z2 , α )δz22 f + Bn (z1 z2 , α )δz2 f + Cn (z1 z2 , α ) f = 0, An (z1 z2 , α )δz21 f + {Bn (z1 z2 , α ) − 2α An (z1 z2 , α )}δz1 f +
(6.25)
Then for m ≥ n we have
where
n = 0, 1, . . . .
= ∂z1 wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
where
um,n = bn (m − n ),
(6.23)
α 2 An (z1 z2 , α ) − α Bn (z1 z2 , α ) + Cn (z1 z2 , α ) f = 0.
fm,n (z1 , z2 ; β|q ) =
z1 m−n z2
n ∂q,z wβ (z1 z2 |q )z2m 2 , wβ ( z1 z2 |q )
(6.33)
(6.18) fm+1,n+1 (z1 , z2 ; β|q ) (6.19)
Similarly, assume that φ n (r; α ) satisfies the second order q-difference equation 2 An (r, α )δq,r f + Bn (r, α )δq,r f + Cn (r, α ) f = 0,
Let fm,n (z1 , z2 ; β|q ) = fm,n (z1 , z2 ; β ), then for m ≥ n we have
(6.20)
=
∂q,z2
m+1−n z2 z1
wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q )
z2 m−n z1
wβ ( z1 z2 |q )
n ∂q,z wβ (z1 z2 |q )z1m 1 fm,n (z1 , z2 ; β|q ) = , wβ ( z1 z2 |q )
,
(6.34)
(6.35)
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fm+1,n+1 (z1 , z2 ; β|q ) =
∂q,z1 wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q ) wβ ( z1 z2 |q )
and
and
∂q,z2
m+1−n z 2
z1
z m−n 2
=
z1
is the little q-Jacobi (α ) Rl,m z, z∗; q2 satisfy α
(6.37)
∂q,z1 wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q ) .
1 2π
k=0
n k!(−1 )k (ν )m+n−k z1m−k z2n−k , k
ν > −1. (7.38)
m,n
( z1 , z2 ) = ( ν )
m n m+n z1 z2 2 F1
−m, −n 1 −m − n − ν + 1 z1 z2
(7.39)
which a constant multiple of the disk polynomials of Section 2.3 in [11]. They have the generating function ∞
um vn = (1 − uz1 − vz2 + uv )−ν , m,n (z1 , z2 ) ( m! n!
Cν
m,n=0
(7.40)
whose proof is an exercise in the application of the binomial theν (z , z ) orem. Next we solve the connection relation between Cm,n 1 2 and Hm, n (z1 , z2 ). We claim that
Cν
m,n
m ∧n
( z1 , z2 ) =
( ν )m+n m! n! Hm−p,n−p (z1 z2 ) 1 p! (m − p)! (n − p)!
p=o
× F1 (−p; −ν − m − n + 1; −1 ) t ν −1
(ν )
0
=
∞
e
(ν )
0 ∞
=
t ν −1
r,s=0
r,s=0
j,k=0
Equating coefficients of um vn we see that m = r + j + k, n = s + j + k. Let p = j + k. Now (7.41) follows after some manipulations. It is clear from the generating function (7.40) that the disk polynomials have the convolution property ν +μ
Cm,n (z1 , z2 ) =
m n
μ
ν C j,k (z1 , z2 )Cm − j,n−k (z1 , z2 )
(7.42)
j=0 k=0
Floris [16], see also [15] and [17], introduced the following qanalogue of the disk polynomials. For α , β > −1 and l, m ∈ Z+ , the (α ) Floris q-disk polynomials Rl,m z, z∗; q2 are defined by (α )
Rl,m z, z∗; q
2
=
α ,l−m )
zl−m Pm(
(α ,m−l )
Pm
1 − zz∗ : q2
∗ m−l
1 − zz : q (z ) ∗
2
l≥m l ≤ m,
where ∗
0
α
∗
α
( ) iθ Rl,m e z, e−iθ z∗; q2 Rl( ,m) eiθ z, e−iθ z∗; q2 dθ
1 − q2
q2 ; q2
1 − q2(α +l+m+1 )
l
q2 ; q2
m
q2m(α +1 ) δl l δmm
q2(α +1 ) ; q2
l
q2(α +1 ) ; q2
, m
(k ) n q 2 (q; q )k (bq; q )m+n−k , (7.43) k (−1 )k z1k−m z2k−n q q k=0
∞ m pm,n (z1 , z2 ; b|q ) = k
it is clear that
pm,n (z2 , z1 ; b|q ) = pn,m (z1 , z2 ; b|q )
(7.44)
then for m ≥ n we have
pm,n (z1 , z2 ; b|q ) = (−1 )n q(2 ) (bq; q )m qm−n+1 ; q z1m−n n n
× pn z1 z2 ; qm−n , b|q ,
(7.45)
where pn (x; a, b|q) is the little Jacobi polynomials. Theorem 7.1. For 0 < q < 1 and b < q−1 , the polynomials { pm,n (z, z; b|q )} satisfy the following orthogonality
pm,n (z, z; b|q ) ps,t (z, z; b|q )dμ(z, z )
(bq; q )∞ qmn (q, bq; q )m (q, bq; q )n δm,s δn,t , 1 − bqm+n+1 (q; q )∞
where
(7.46)
(bq; q ) qk dθ k δ r − qk/2 , 2π (q; q )k ∞
(7.47)
z = reiθ , r ∈ R+ , θ ∈ [0, 2π ] and m, n, s, t ∈ N0 .
ur vs r+s Hr,s (z1 z2 ) t dt r! s!
∞ ur vs (−1 ) j (uv ) j+k (ν + r + s + j + 2k ) . r! s! j! k! (ν )
Hr,s (z1 z2 )
polynomials
k=0
∞
−t +t (t −1 )uv
0
2π
dμ(z, z ) =
e−t +t uz1 +t vz2 −t uvdt
q-disk
for α We now introduce our q-analogue of these polynomials. For 0 < q < 1 and b < q−1 , let us define
= (7.41)
The
> −1, l , l , m, m ∈ Z+ .
C
Write the right-hand side of (7.40) as ∞
1
They are also known as the disk polynomials or the Zernike polynomials, [11]. It is clear that
Cν
=
m k
α
( ) = Rm,l z, z∗; q2
× (1 − zz∗ )α dq2 (1 − zz∗ )
The 2D-ultraspherical polynomials are ν (z , z ) = Cm,n 1 2
∗
polynomials.
and the orthogonality relation
7. 2D q-ultraspherical polynomials
m ∧n
( ) Rl,m z, z∗; q2
wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q )
( x : q ) = pm x; qα , qβ ; q
α ,β )
Pm( (6.36)
[m5G;January 27, 2017;14:59]
Theorem 7.2. The polynomials {pm, n (z1 , z2 ; b|q)} have the generating function ∞ pm,n (z1 , z2 ; b|q ) m n u v (q; q )m (q; q )n
m,n=0
uz1 , vz2 (bq, uv; q )∞ = 2 φ1 uz , z v ; q uv ( 1 2 )∞
q; bq .
(7.48)
For bq, cq < 1 and b = 0, the connection relation between the q − 2D ultraspherical polynomials is given by
pm,n (z1 , z2 ; b|q ) = (bq; q )∞
∞ j=0
c
;q b
j
( q; q ) j
bq
m+n 2 +1
j
pm,n z1 q 2 , z2 q 2 ; c q j
j
(cq; q )∞
.
(7.49) The connection relation between the q − 2D ultraspherical and q − 2D Hermite is given by
∞ (m+n )/2+1 pm,n (z1 , z2 ; b|q ) bq = (bq; q )∞ ( q; q ) j
j
Hm,n z1 q j/2 , z2 q j/2 |q .
j=0
2
∗
2
z z = q zz + 1 − q ,
(7.50)
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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Moreover we have the inverse relation
Hm,n (z1 , z2 |q ) =
∞ −bq(m+n )/2+1 k=0
k
(bq; q )∞ (q; q )k
q(2 ) pm,n z1 qk/2 , z2 qk/2 ; b|q . k
(7.51)
2
φ1 =
uz1 , vz2 uv
1 − qm−n pm+1,n+1 (z1 , z2 ; b|q )
= z1 1 − bqm+1 − z2 q
m−n
− z1 1 − bq
∞ (uz1 , z2 v; q )∞ pm,n (z1 , z2 ; b|q ) m n u v . (bq, uv; q )∞ m,n=0 (q; q )m (q; q )n
(7.52)
1 − qm+1 pm,n+1 (z1 , z2 ; b|q )
n+1
1−q
n+1
(7.64)
pm+1,n (z1 , z2 ; b|q ),
pm+1,n+1 (z1 q, z2 ; b|q ) − pm+1,n+1 (z1 , z2 q; b|q ) = z2 1 − bqm+2
q; bq
1 − bq
Let us rewrite (7.48) in the form
13
m+1
1 − qn+1 pm+1,n (z1 q, z2 ; b|q )
1−q
m+1
(7.65)
pm,n+1 (z1 , z2 q; b|q ),
z2 (1 − qn ) pm,n−1 (z1 q, z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 q; b|q ) = z2 (1 − qn ) pm,n−1 (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ), (7.66)
Theorem 7.3. The polynomials {pm, n (z1 , z2 ; b|q)} satisfy the following properties,
Dq,z1 pm,n (z1 , z2 ; b|q ) =
(1 − bq ) 1−q
(1 − qm ) pm−1,n (z1 , z2 ; bq|q ),
qz1 z2 ( pm,n (z1 q, z2 ; b|q ) − pm,n (z1 , z2 q; b|q )) + z1 z2 (1 − qm )(1 − qn )( pm−1,n−1 (z1 q, z2 ; b|q ) − pm−1,n−1 (z1 , z2 q; b|q ))
(7.53) Dq,z2 pm,n (z1 , z2 ; b|q ) =
(1 − bq ) 1−q
(1 − q ) pm,n−1 (z1 , z2 ; bq|q ), n
(7.54)
(qz1 z2 ; q )∞ Dq−1 ,z1 pm,n (z1 , z2 ; b|q ) (bqz1 z2 ; q )∞ (qz1 z2 ; q )∞ pm,n+1 z1 , z2 ; bq−1 |q = , qm−1 (q − 1 )(bz1 z2 ; q )∞ (qz1 z2 ; q )∞ Dq−1 ,z2 pm,n (z1 , z2 ; b|q ) (bqz1 z2 ; q )∞ (qz1 z2 ; q )∞ pm+1,n z1 , z2 ; bq−1 |q = , qn−1 (q − 1 )(bz1 z2 ; q )∞
(7.56)
(7.57) pm,n (z1 q, z2 ; b|q ) − bq2m−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 q; b|q ) = pm,n (z1 , z2 ; b|q ) − qm−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ), (7.58) pm,n (z1 , z2 q; b|q ) − bq
(1 − q )(1 − q ) pm−1,n−1 (z1 q, z2 ; b|q ) n
= qn pm,n (z1 , z2 ; b|q ) − qn−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ), (7.59) pm,n (z1 , z2 q; b|q ) − bqm+n (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 q; b|q ) = qn pm,n (z1 , z2 ; b|q ) − qn−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ). (7.60) pm,n (z1 q, z2 ; b|q ) − bqn+1 z1 (1 − qm ) pm−1,n (z1 q, z2 ; b|q ) = pm,n (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ), pm,n (z1 q, z2 ; b|q ) − bqm z1 (1 − qm ) pm−1,n (z1 , z2 q; b|q ) = pm,n (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ),
+ z1 (1 − qm )( pm−1,n (z1 q, z2 ; b|q ) − pm−1,n (z1 q, z2 q; b|q )) − z2 (1 − qn )( pm,n−1 (z1 , z2 q; b|q ) − pm,n−1 (z1 q, z2 q; b|q )). The inversion transformation of quanta q → q−1 in (4.6) relates the properties of one family of polynomials for q > 1 to the properties of another family of polynomials with 0 < q < 1. The polynomials pm, n (z1 , z2 ; b|q) are essentially invariant under the quanta inversion transformation,
pm,n z1 , z2 ; b|q−1
= pm,n (z1 , z2 ; b|q )qm − qm−1 (1−qm )(1−qn ) pm−1,n−1 (z1 , z2 ; b|q ),
m
− qz12 z2 (1 − qm )( pm−1,n (z1 , z2 q; b|q ) − bpm−1,n (z1 q, z2 q; b|q ))
(7.67) (7.55)
pm,n (z1 q, z2 ; b|q ) − bqn+m−1 (1−qm )(1−qn ) pm−1,n−1 (z1 q, z2 ; b|q )
2n−1
= qz1 z22 (1 − qn )( pm,n−1 (z1 q, z2 ; b|q ) − bpm,n−1 (z1 q, z2 q; b|q ))
m+n
= (−b)
∞ m k k=0
q
n k
(−b)−k (q; q )k
m+n+1 q ( 2 )
m+n −
= (−b) ×
q b
;q
∞
k+1 2
m+n−k
;q
m+n−k
) zm−k zn−k 1 2 q k k n q (2 ) − ( q; q )
m+n−k+1 2
k=0
b
q
×qk(k−m )+k(k−n )−( )−( )−( k 2
q
m k
q
k
k
b
q
z1m−k z2n−k
m+n q ( 2 ) pm,n 1 α 1 −α × (b/q ) z1 , (b/q ) z2 ; |q . b
= (−1 )m+n (b/q )(
1−α )m+α n −
Therefore, we have established the symmetry
pm,n z1 , z2 ; b|q−1
=
bq−1
(1−α )m+αn m+n
(−1 )
q(
m+n 2
)
α
1 −α
pm,n (b/q ) z1 , (b/q )
z2 ; 1/b|q ,
(7.68)
(7.61)
for α ∈ C. We now come the asymptotics of pm,n (z1 , z2 ; b|q).
(7.62)
Theorem 7.4. Let z1 , z2 ∈ C, bq < 1 and z1 z2 = 0, then we have
lim
m,n→∞
1 + bqm+n z1 pm,n (z1 , z2 ; b|q ) = z1 pm−1,n (z1 , z2 ; b|q ) − qm−1 (1 − qn )(1 − bqn ) pm−1,n−1
pm,n (z1 , z2 ; b|q ) = (bq; q )∞ z1m z2n
1 z1 z2
;q
∞
,
(7.69)
uniformly on compact subsets of the z1 and z2 planes.
× (z1 , z2 ; b|q ), (7.63)
The theorem follows from the definition (7.43) and Tannery’s theorem.
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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(β )
8. The polynomials {Zm,n (z1 , z2 )}
(β ) ) (nδz1 − mδz2 )Zm,n (z1 , z2 ) = (m − n )(m + β )Zm(β−1 ( z , z ), ,n−1 1 2
Motivated by the class of general 1D systems in Section 2 we (β ) define polynomials {Zm,n (z1 , z2 )} by
n 1 n ( β + 1 )m (β ) Zm,n (z1 , z2 ) = (−1 )n−k z1m−k z2n−k , n! k (β + 1 )m−k
(8.1)
k=0
for m ≥ n. When m < n the polynomials are defined by (β )
(β )
Zm,n (z1 , z2 ) = Zn,m (z2 , z1 )
Here β > −1. These polynomials arise through the choice φn (x; α ) = ( α +β ) Ln (x ), where Ln(a) (x ) is a Laguerre polynomial, [25,50], ) n n Ln(a ) (x ) = (a+1 k=0 n!
(−x )k k (a+1 )k
)n n = (a+1 k=0 n!
n
(−x )n−k . k (a+1 )n−k
Zm,n (z1 , z2 ) =
(β +m−n ) z1m−n Ln
( z1 z2 )
z1 Zm,n
(8.3)
(β )
(z1 , z2 ) = Zm+1,n (z1 , z2 ).
(α + β + n + 1 )
(8.6)
(β )
Zm,n (z, z¯ ) Zs,t (z, z¯ ) (zz¯ )β e−zz¯ rdrdθ
(β )
(8.8)
(β +1 )
) (z1 , z2 ) = −Zm(β+1 (z , z ) ,n 1 2
(8.20)
1 (β ) (β ) ∂z wβ (z1 z2 )Zm,n (z1 , z2 ) = (n + 1 )Zm,n (z , z ) +1 1 2 wβ ( z1 z2 ) 1
δz j wβ (z1 z2 ) = (β − z1 z2 )wβ (z1 z2 ),
j = 1, 2
(8.22)
and (8.8) which imply the relationships
(β ) δz2 wβ (z1 z2 )Zm,n ( z1 , z2 )
(β )
(β )
= β Zm,n (z1 , z2 ) − z2 Zm+1,n (z1 , z2 ), (8.23)
wβ ( z1 z2 ) (β )
z2 Zm+1,n (z1 , z2 ) = −(n + 1 )Zm+1,n+1 (z1 , z2 )
(β )
(8.24)
(β )
(β )
+ (β + m + 1 )Zm,n (z1 , z2 ),
(8.9)
and
Theorem 8.3. The polynomials {Zm,n (z1 , z2 )} satisfy the second order partial differential equation
∂∂ z1 ∂∂ z2 f +
(β ) ( z1 , z2 ) (β + m + n + 1 − z1 z2 )Zm,n (β )
(β ) δz1 wβ (z1 z2 )Zm,n ( z1 , z2 ) = (β + m − n )Zm,n (z1 , z2 ) − z2 Zm+1,n (z1 , z2 ).
(β )
(β )
= (n + 1 )Zm+1,n+1 (z1 , z2 ) + (m + β )Zm−1,n−1 (z1 , z2 )
(8.10)
respectively, where m ≥ n. We now discuss differential recurrence relations. (β )
Theorem 8.2. For m ≥ n, the polynomials {Zm,n (z1 , z2 )} satisfy the differential recurrence relations (β ) (β ) δz2 Zm,n (z1 , z2 ) = −z2 Zm,n ( z , z ), −1 1 2
(8.11)
(β ) (β ) ) δz2 Zm,n (z1 , z2 ) = nZm,n (z1 , z2 ) − (β + m )Zm(β−1 ( z , z ), ,n−1 1 2
(8.12)
(β ) (β ) (β ) δz1 Zm,n (z1 , z2 ) = (m − n )Zm,n (z1 , z2 ) − z2 Zm,n ( z , z ), −1 1 2
(8.13)
(β ) (β ) ) δz1 Zm,n (z1 , z2 ) = mZm,n (z1 , z2 ) − (m + β )Zm(β−1 (z , z ), (8.14) ,n−1 1 2 (β )
(β )
and
z1 Zm,n (z1 , z2 ) = Zm+1,n (z1 , z2 ) − Zm,n−1 (z1 , z2 ), (β )
(8.7)
In view of (8.6) the recurrence relations (6.8), (6.7) and (6.13) become (β )
(8.19)
It is clear that (8.18) implies the differentiation formulas
wβ ( z1 z2 )
(β + m ∨ n + 1 ) δm,s δn,t . ( m ∧ n )!
(β )
(8.18)
for m ≥ n. It also clear that
Theorem 8.1. For m, n, s, t = 0, 1, . . . and β > −1, we have the orthogonality relation
=π
(8.21)
, n! n− j (−1 ) (α + β + 1 )n c j (n, α ) = . ( n − j )! j ! ( α + β + 1 ) n − j
(β )
(8.17)
2
1 −β z1 z2 n m+β −z1 z2 z e ∂z1 z 1 e n! 1 (−1 )m = (z1 z2 )−β ez1 z2 ∂zn1 (z1 z2 )β ∂zm2 e−z1 z2 . n!
(β )
Zm,n (z1 , z2 ) =
and
(8.5)
Therefore in the notation of Section 2, we have
R2
β +m n z
(8.4)
(0 ) Hm,n (z1 , z2 ) = (−1 )n (m ∧ n )!Zm,n ( z1 , z2 ).
−β
z1 exp(−∂z1 ∂z2 )z1
ez1 z2 ∂z2 e−z1 z2 Zm,n (z1 , z2 ) = −z1 Zm,n
It is clear that when β = 0 we see that
ζn (α ) =
n!
wβ (z1 z2 ) = (z1 z2 )β exp(−z1 z2 ).
and (β +1 )
(−1 )n
(β )
Zm,n (z1 , z2 ) =
We shall use the notation
Indeed, for m ≥ n (β )
where δz1 = z1 ∂z1 , δz2 = z2 ∂z2 . Moreover they have the operational representation
and the Rodrigues type representation
(8.2)
n
(8.16)
(β )
(δz1 − δz2 )Zm,n (z1 , z2 ) = (m − n )Zm,n (z1 , z2 ),
(8.15)
β − z1 z2 z1
∂∂ z2 f = −n f,
(8.25)
for all m ≥ n. It is clear that the differential property (8.25) can be written in the form
∂ ∂ (β ) (β ) w (z z ) Z (z , z ) = −nZm,n (z1 , z2 ). ∂ z1 β 1 2 ∂ z2 m,n 1 2
(8.26)
Note that (8.25) and (8.15) indicate that the polynomials (β ) {Zm,n (z1 , z2 )} are simultaneous eigenfunctions of the operators on their respective left-hand sides. The differential operators β −z1 z2
∂∂ z1 ∂∂ z2 +
z1
∂∂ z2 and z1 ∂∂ z1 − z2 ∂∂ z2 indeed commute as
can be directly verified. Let w(x, y ) = (zz¯ )β e−zz¯ and define the inner product
< f, g >=
R2
f (x, y )g(x, y )w(x, y )dxdy.
(8.27)
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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( β ,γ )
9. The polynomials {Mn
Theorem 8.4. We have the adjoint relations
(∂z )∗ = −∂z¯ −
β
+ z,
z¯
(∂z¯ )∗ = −∂z −
β z
+ z¯.
(8.28)
The proof is straightforward calculus exercise. This allows us to write (8.25) or (8.26) in the selfadjoint form
( (∂z¯ )∗ ∂z¯ ) f = n f.
15
( z1 , z2 )}
For α > 0, β , γ > −1 Ismail and Zeng [29] introduced the poly( α + γ ,β ) nomials φn (r; α ) = Pn (1 − 2r ), that is
φn (r; α ) = (α + γ + 1 )n
(8.29)
n (α + β + γ + n + 1 )n−k (−r )n−k . k!(n − k )!(α + γ + 1 )n−k k=0
(9.1)
Moreover we also have the following result. Theorem 8.5. The operator A := (∂z¯ )∗ ∂z¯ is positive in the sense that < Af, f > ≥0 with = if and only if ∂z¯ f = 0, that is f depends only on z.
They satisfy the following orthogonality relation
1 0
φm (r; α )φn (r; α )uα+γ (1 − u )β du = ζn (α + γ , β ) δm,n , (9.2)
where
This indicates that
1 ∂ (β ) (β ) w(x, y )Zm,n (z, z¯ ) = (n + 1 )Zm,n (z, z¯ ). w(x, y ) ∂ z
(8.30)
ζn (α + γ , β ) =
(α + γ + n + 1 )(β + n + 1 ) . n!(α + β + γ + n + 1 )(α + β + γ + 2n + 1 ) (9.3)
This is the adjoint of (8.11). Ismail and Zeng [29] established the connection relations stated in the next theorem.
We define the two variable polynomial Mm,n (z1 , z2 ) by
Theorem 8.6. We have the connection relation
Mm,n (z1 , z2 ) =
(β ,γ )
(9.4)
( m ≥ n ).
for m ≥ n and
j=0
(8.31) For m ≥ n, we have the special cases
n (−β ) j Hm,n (z1 , z2 ) = n! (−1 ) j Zm(β−)j,n− j (z1 , z2 ). j!
(8.34) (β )
(8.35)
(8.36) It is clear that the generating function (8.36) implies the identity
(β )
(γ )
Z j,k (z1 , z2 )Zm− j,n−k (z3 , z4 )
m≥ j≥k≥0
( j − k )! ( m − n − j + k )!
.
(8.37)
(α +β +1 )
(x + y ) =
n (β ) Lk(α ) (x )Ln (y ).
(−1 )n n! (β ,0) M (z, z¯ ), (β + 1 )n m,n
m ≥ n.
(9.8)
From (9.1) we get
ck (n, α ) =
(m + β + γ + 1 )n−k (γ + 1 )m (−1 )n−k , k!(n − k )!(γ + 1 )m−k
(9.9)
where α = m − n with m ≥ n. Then ,γ ) (z , z ) (β + γ + m + n + 2 )z2 Mm(β+1 ,n 1 2
(β ,γ )
(β ,γ )
= (γ + m + 1 )Mm,n (z1 , z2 ) − (n + 1 )Mm+1,n+1 (z1 , z2 ), (β ,γ ) ( z1 , z2 ) (β + γ + m + n + 1 )z1 Mm,n (β ,γ )
(β ,γ )
= (β + γ + m + 1 )Mm+1,n (z1 , z2 ) − (β + n )Mm,n−1 (z1 , z2 )
(9.10)
(9.11)
and (β ,γ ) ,γ ) ,γ ) (cn −z1 z2 )Mm,n (z1 , z2 )=an Mm(β+1 (z , z ) + bn Mm(β−1 ( z , z ), ,n+1 1 2 ,n−1 1 2
This is an analogue of the convolution identity
Ln
(9.7)
for m ≥ n as it is stated in [29]. The disk polynomials defined in [11] can be expressed as β
uvz z + z u um vn 1 2 1 (β ) Zm,n (z1 , z2 ) = (1 + uv )−β −1 exp . ( m − n )! 1 + uv m≥n≥0
( z1 + z3 , z2 + z4 ) =
(9.6)
(γ + m + 1 ) (β + n + 1 ) δm,p δn,q , n!(β + γ + m + 1 )(β + γ + m + n + 1 )
Pm,n (z ) =
∂ Zm,n (β ) + nz1 Zm,n = 0. ∂ z2
,γ ) (z1 , z2 ) = Mm(β+1 ( z , z ). ,n 1 2
(β ,γ ) (β ,γ ) dθ 2γ 2 β |z|≤1 Mm,n (z, z¯ )M p,q (z, z¯ )r (1 − r ) rdr π
=
Ismail and Zeng [29] gave the generating function
(β +γ +1 )
(9.5)
( β ,γ )
(8.33)
Therefore Theorem 6.4 shows that
Zm,n
z1 Mm,n
The Laguerre differential equation is [45,50],
xy + (1 + α − x )y + ny = 0.
∂
m < n.
Clearly, Mm,n (z1 , z2 ) satisfy the orthogonality relation
j=0
+ ( 1 + β + m − n − z1 z2 )
(β ,γ )
(β ,γ +1 )
(8.32)
j=0
m,n z22
(β ,γ )
Mm,n (z1 , z2 ) = Mn,m (z2 , z1 ), Then,
n n (β ) n!Zm,n (z1 , z2 ) = (β ) j (−1 ) j Hm− j,n− j (z1 , z2 ), j
z2
n (m + β + γ + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k k!(n − k )!(γ + 1 )m−k k=0
n (β − γ ) j Zm,n (z1 , z2 ) = (−1 ) j Zm(γ−)j,n− j (z1 , z2 ) j!
(β )
∂ 2 Z (β )
β ,γ
(9.12) (8.38)
k=0
Problem. Al-Salam and Chihara characterized all 1-D orthogonal polynomials satisfying formulas of convolution type in [2]. They discovered the Al-Salam–Chihara polynomials through this characterization, [25]. It will be interesting to solve the corresponding 2D characterization problem.
where
an =
(n + 1 )(β + γ + m + 1 ) , (β + γ + m + n + 1 )(β + γ + m + n + 2 )
(9.13)
bn =
(γ + m )(β + n ) , (β + γ + m + n )(β + γ + m + n + 1 )
(9.14)
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n (γ + m ) (n + 1 )(γ + m + 1 ) − . β +γ +m+n+2 β +γ +m+n
cn =
=n
(9.15)
k=0
It must be noted that (9.12) is essentially the three term recurrence relation for Jacobi polynomials [50]. It is clear that
δ φn (x, α ) + Bn δx φn (x, α ) + Cn φn (x, α ) = 0,
An x2
n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k − (k − 1 )!(n − k )!(γ + 1 )m−k k=1
(β ,γ )
An = 1 − x,
(9.17)
α + γ − x(α + β + γ + 1 ),
(9.18)
Similarly, Ismail and Zeng [29] established the following generating functions: ∞
(β ,γ )
Mm,n (z1 , z2 )um vn
m,n=0
Cn = xn(α + β + γ + n + 1 ).
(9.19)
Then for m ≥ n, Mm,n (z1 , z2 ) satisfy the following second order partial differential equation
(1 − z1 z2 )δz22 f (z1 , z2 ) + {m − n + γ − z1 z2 (β + γ + m − n + 1 )}δz2 f (z1 , z2 ) + z1 z2 n ( β + γ + m + 1 ) f ( z1 , z2 ) = 0.
2β +γ
=
( β ,γ )
×
ρ
(9.20)
( 1 + u v + ρ ) −β ( 1 − u v + ρ ) −γ
1 1 + −1 , 1 − 2z1 u/(1 + ρ − uv ) 1 − 2vz2 /(1 + ρ − uv ) (9.23)
and
(β ,γ )
Mm,n (z1 , z2 )
An equivalent form is
( 1 − z1 z2 )z2
(β +1,γ )
= nMm,n (z1 , z2 ) − (γ + m )Mm−1,n−1 (z1 , z2 ).
(9.16)
where
Bn =
n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k k!(n − k )!(γ + 1 )m−k
m≥n≥0
∂ 2 f ( z1 , z2 ) ∂ z22
2β +γ
=
+ [1 + m − n + γ − z 1 z 2 ( 2 + γ + β + m − n ) ] +z1 n ( m + β + γ + 1 ) f ( z1 , z2 ) = 0.
ρ
∂ f ( z1 , z2 ) ∂ z2
× exp (9.21)
Similarly we establish
um vn ( m − n )!
( 1 + u v + ρ ) −β ( 1 − u v + ρ ) −γ 2z u 1
1 − uv + ρ
,
where
ρ = ( 1 − 2 u v ( 1 − 2 z 1 z 2 ) + u 2 v2 ) 1/2
∂ 2 f ( z1 , z1 ) ( 1 − z1 z2 )z1 ∂ z21
(9.24)
(9.25)
They also verified the limiting relation
+ [1 − m + n + γ − z 1 z 2 ( 2 + γ + β − m + n ) ] + z2 m ( n + β + γ + 1 ) f ( z1 , z2 ) = 0.
(β ,γ )
β →∞
(9.22)
Theorem 9.1. For m ≥ n ≥ 0, β > −1, γ > −1 we have
∂ (β ,γ ) (β +1,γ ) M (z , z ) = −(β + γ + m + 1 )Mm,n ( z1 , z2 ) ∂ z2 m,n 1 2 (β ,γ ) (β ,γ ) +1,γ ) δz2 Mm,n (z1 , z2 ) = nMm,n (z1 , z2 ) − (γ + m )Mm(β−1 (z , z ) ,n−1 1 2 (β +1,γ ) (β ,γ ) = (β + γ + m + 1 ) Mm,n (z1 , z2 ) − Mm,n ( z1 , z2 ) , = (β + γ + m + 1 )Mm,n
( z1 , z2 )
(β ,γ )
m≥n≥0
(β ,γ )
−(β + γ + n + 1 )δz2 Mm,n (z1 , z2 ) = (m − n )(β + γ + m + 1 )Mm,n
( z1 , z2 ),
(β ,γ ) (β ,γ ) (z1 , z2 ) = (m − n )Mm,n ( z1 , z2 ), (δz1 − δz2 )Mm,n (β ,γ )
(β + γ + m + 1 )δz1 Mm,n (z1 , z2 ) (β ,γ )
−(β + γ + n + 1 )δz2 Mm,n (z1 , z2 ) (β +1,γ )
= (m − n )(β + γ + m + 1 )Mm,n
2β +γ (1 + uv + ρ )−β (1 − uv + ρ )1−γ , ρ (1 − uv − 2uz1 + ρ ) (9.26)
with ρ as in (9.25). 10. q-Laguerre type 2 − D polynomials For α > −1 the q-Laguerre polynomials n q(α +n−k )(n−k ) (−x )n−k , (q; q )k (q, qα+1 ; q )n−k
(10.1)
k=0
(β ,γ ) ( z1 , z2 ) (β + γ + m + 1 )δz1 Mm,n (β +1,γ )
(β ,γ )
um vn Mm,n (z1 , z2 )=
Ln(α ) (x; q ) = (qα +1 ; q )n
−(β + γ + n + 1 )Mm,n (z1 , z2 ),
(γ ) (z1 z2 ) = (−1 )n Zm,n ( z1 , z2 ),
and found the generating relation
(β ,γ ) (β ,γ ) +1,γ ) δz1 Mm,n (z1 , z2 ) = mMm,n (z1 , z2 ) − (γ + m )Mm(β−1 (z , z ) ,n−1 1 2 (β +1,γ )
(γ +m−n )
lim Mm,n (z1 , z2 /β ) = z1m−n Ln
∂ f ( z1 , z2 ) ∂ z2
( z1 , z2 ).
and (β ,γ ) δz2 Mm,n ( z1 , z2 ) n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k (n − k ) = k!(n − k )!(γ + 1 )m−k k=0
are orthogonal to infinitely many measures. Some orthogonality measures are
0
∞
(α ) Lm (x; q )Ln(α ) (x; q )xα dμ(x|q ) = ζn (α )δm,n ,
(10.2)
are given by
d μ ( x|q ) =
xα d x
(−x; q )∞
,
(q−α ; q )∞ (qα+1 ; q )n , ( q; q )∞ qn ( q; q )n ∞ xδ (x − cqk ) d μ ( x|q ) = , (−x; q )∞ k=−∞
ζn (α ) = (−α )(α + 1 )
(10.3)
−α
(q, −cqα+1 , − q c ; q )∞ (qα+1 ; q )n ζn (α ) = α+1 (q , −c, − qc ; q )∞ c−α−1 (q; q )n qn
(10.4)
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and
d μ ( x|q ) =
ζn (α ) =
x
−c
−λx, − λqx ; q
dx
∞ , −x, − qc x, − λqqc x ; q ∞ ∞ xα −c − x, − q ; q dx λx ∞ , −x, − qc x, − λqqc x ; q 0 ∞
λ
zm,n (z1 , z2 |q ) =
(β +m−n )
z1m−n Ln
1 − qβ
= (10.5)
where c, λ > 0. Following the procedure outline in Sections 6 we let (β )
(z1 z2 ; q ), m ≥ n,
(β )
zn,m (z2 , z1 |q ),
(10.6)
m < n,
1−q
(β )
zm,n (z1 , z2 |q ) =
k=0
(−1 )
wβ ( z1 z2 ; q )
n−k
( q; q )k ( q; q )
m−k n−k z2 m z1 . β +1 ; q n−k q m−k
(
)
(10.7)
(β )
z1 zm,n (z1 , z2 |β ) = zm+1,n (z1 , z2 |β ), (0 ) zm,n ( z1 , z2 |q ) =
=
(β )
Dq,z2 z2m−n wβ (z1 z2 ; q )zm,n (z1 , z2 )
1 − qn+1 (β ) z (z1 , z2 ), (10.18) 1 − q m,n+1
=
( z1 z2 ; q )
(−1 )n h ( z , z |q ), (q; q )n m,n 1 2
m ≥ n.
where m > n.
(10.8)
(β )
+
(10.9)
for m, n, s, t = 0, 1, . . . and dμ(x|q), ζ n (α ) may be any pair from (10.3), (10.4) or (10.5).
where θq,z = zDq,z has a solution y(x ) = Ln(α ) (x; q ). This leads to:
2 (1 + qz1 z2 )θq,z f+ 2
+
In the case
(10.10)
+
Theorem 10.2. For β > −1 and m ≥ n we have (β )
q(m+1+β ) z2 zm+1,n (z1 , z2 ) = (qn+1 − 1 )zm+1,n+1 (z1 , z2 ) (10.11)
(β )
qn z1 zm,n+1 (z1 , z2 ) = zm+1,n (z1 , z2 ) − zm,n−1 (z1 , z2 ) 1 + q ( 1 − qn − qm+β ) − z1 z2 q = (1 − q
1+n
(10.12)
pn ( x; qα |q ) =
(k−n )(n+k−1 ) n 2 ( q; q )n q (−x )n−k α +1 (q; q )k (q, q ; q )n−k
(β )
(β )
zm,n (z1 , z2 )
(β )
(β ) )zm+1,n+1 (z1 , z2 ) + q(1 − qβ +m )zm ( z , z ). −1,n−1 1 2
(β )
wm,n (z1 , z2 |q ) =
z1m−n pn (z1 z2 ; qβ +m−n |q ), m ≥ n, (β )
wn,m (z2 , z1 |q ),
for β > −1 and m, n = 0, 1, . . .. Then for m ≥ n find that (β +1 )
(β )
z1 wm,n (z1 , z2 |β ) = wm+1,n (z1 , z2 |β ),
Theorem 10.3. For β > −1 let wβ (x; q ) = xβ /(−x; q )∞ . Then we have
wm,n (z1 , z2 |q ) =
(β )
z1n−m (β )
Dq,z2 zm,n (z1 , z2 ) =
(β )
zm,n (z1 , z2 ) =
=
(1 − q )m+n Dnq,z1
β z Dm 2
q,z2
−β z2 wβ
(−1 ) (q; q )n wβ (z1 z2 ) m
(10.14)
(10.15)
( z1 z2 )
(10.25)
n (q; q )n (qβ +1 ; q )m−n z1m−k z2n−k q−(2 )−(2 ) k
(β )
k=0
n
(−1 )n−k (q; q )k (q; q )n−k (qβ +1 ; q )m−k
. (10.26)
qβ z2 ( β ) z (qz1 , z2 ), q − 1 z1 m,n−1
qβ ( β ) z (qz1 , z2 ), q − 1 m,n−1
(10.24)
m < n,
The raising and lowering relations of the q-Laguerre polynomials lead to the following q-difference relations,
Dq,z1 z1n−m zm,n (z1 , z2 )
(10.23)
k=0
we define the 2D polynomials wm,n (z1 , z2 |q ) through
β +m+n+1
(10.13)
(10.22)
q ( 1 − qβ +m )z1 z2 f = 0. ( 1 − q )2
Starting with the little q-Laguerre [38] or Wall polynomials
(β )
and
1 − qβ +m−n + (2 − qβ +m+1 )qz1 z2 θq,z1 f 1−q
[9]
+ (1 − qm+β +1 )zm,n (z1 , z2 ), (β )
(10.21)
qz1 z2 (1 − qn ) f =0 ( 1 − q )2
2 (1 + qz1 z2 )θq,z f− 1
Applying formulas (6.7) and (6.8) lead to the following theorem.
(β )
qn−m−β − 1 − (2 − qn )qz1 z2 θq,z2 f 1−q
and
(qα+1 ; q )n q(α+n−k)(n−k) (−1 )n−k (q; q )k (q, qα+1 ; q )n−k
(β )
(10.20)
qx(1 − qn ) y ( x ) = 0. ( 1 − q )2
(β )
= ζm∧n (|m − n| + β )δm,s δn,t ,
ck (n, α ) =
1 − qα − 2qα +1 x + qα +n+1 x θq,x y(x ) qα ( 1 − q )
Theorem 10.4. For m ≥ n the function f = zm,n (z1 , z2 ) satisfies the second order q-difference equations
(β )
zm,n (z, z¯|q )zs,t (z, z¯|q )r 2β dμ(r 2 |q )dθ
1 − qn+1 z1 (β ) z ( z1 , z2 ), 1 − q z2 m,n+1 (10.19)
2 (1 + qx )θq,x y (x ) +
Theorem 10.1. Let β > −1. Then the following orthogonality relations
(10.17)
The following second order q-difference equation
(β +1 )
R2
(β )
z2m−n wβ q(β +m−k )(n−k ) (qβ +1 ; q )
qβ z2 ( β ) z ( z1 , z2 ), 1 − q m+1,n
Dq,z1 wβ (z1 z2 ; q )zm,n (z1 , z2 )
Therefore
(β )
zm,n (z1 , z2 ) −
where m ≥ n and
and use the notation n
wβ ( z1 z2 )
λ λ
(β )
z2 Dq,z2 wβ (z1 z2 )zm,n (z1 , z2 )
17
The orthogonality relation for the little q-Laguerre polynomials is ∞
q(α +1)k (qk+1 ; q )∞ pm (qk ; qα |q ) pn (qk ; qα |q )
k=0
,
(10.16)
(q; q )∞ q(α+1)n (q; q )n δm,n = , (qα+1 ; q )∞ (qα+1 ; q )n
(10.27) m, n = 0, 1, . . . .
This leads to the orthogonality relation below.
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Theorem 10.5. For m, n, s, t = 0, 1, . . . and β > −1 we have the following orthogonality relations
(β )
(β )
wm,n (z, z¯|q )ws,t (z, z¯|q )r 2β dμ(r 2 |q )dθ
R2
(β )
=
β +m−n )w ( z , z |q ) z1 ( 1 − q m,n+1 1 2 z2 qβ +m−n−1 (1 − q )
and
= ζm∧n (|m − n| + β )δm,s δn,t ,
(10.28)
(β )
z2 Dq−1 ,z2 w(z1 z2 ; β|q )wm,n (z1 , z2 |q )
where z = reiθ ,
d μ ( r |q ) =
∞
(β )
x(xq; q )∞ δ (x − qk ), (10.29)
The function y(x ) = pn (x; qα |q ) satisfies the q-difference equation 2 qα +n−1 θq,z y (z ) +
q
n−1
+ qα +n−1 − z 1−q
qβ +1−n 1 − qβ +m+1 1 − qβ ( β ) (β ) wm,n (z1 , z2 |q )− z2 wm+1,n (z1 , z2 |q ). 1−q 1 − q 1−qβ +m−n+1 (10.39)
=
k=0
θq,z y(z ) +
z (1 − q ) y (z ) = 0 ( 1 − q )2 (10.30) n
Applying the procedure of Section 6 to (k−n )(n+k−1 )
ck (n, α ) =
(β )
Theorem 10.6. For β > −1, m ≥ n the polynomial wm,n (z1 , z2 |q ), satisfies the q-partial difference equations
qn−1 − qβ +m−1 − z1 z2 ( 1 − qn )z1 z2 θq,z2 f + f = 0, q,z2 f + 1−q ( 1 − q )2 (10.31) qn − qβ +m + qz1 z2 q ( 1 − qβ +m )z1 z2 θq,z1 f + f = 0. 1−q ( 1 − q )2 (10.32)
Theorem 10.8. For β > −1, m ≥ n we have (β )
qn (1 − qm−n+β +1 )
z1 ( 1 − q
The following theorem follows from the raising and lowering relations of the little q-Laguerre polynomials. Theorem 10.7. For β > −1 and m ≥ n we have (β )
Dq,z1 z1n−m wm,n (z1 , z2 |q )
z1n−m
(β )
=−
1−n n z2 q (1 − q )wm,n−1 (z1 , z2 |q ) , z1 (1 − q )(1 − qβ +m−n+1 )
(β )
Dq,z2 wm,n (z1 , z2 |q ) = −
(β )
q1−n (1 − qn )wm,n−1 (z1 , z2 |q )
(1 − q )(1 − qβ +m−n+1 )
,
(10.35)
= (1 − q −q
m−n+β +1
qm(n−1)+n(β −1)−(2 ) (1 − q )m+n (−1 )m (qβ +m−n+1 ; q )n
−β × Dnq−1 ,z1 (z1 z2 )β Dm w(z1 z2 ; β|q ) q−1 ,z2 (z1 z2 )
. (10.36)
For m > n we have
(β )
Dq−1 ,z1 w(z1 z2 ; β|q )wm,n (z1 , z2 |q )
Dq−1 ,z2
(β )
(z1 z2 ; β|q )wm,n (z1 , z2 |q )
z2m−n w
(1 − q )wm,n−1 (z1 , z2 |q ),
(β )
qn + qm+β (1 − qn − qn+1 ) − z1 z2 wm,n (z1 , z2 |q ) (β )
= qn (1 − qm+β +1 )wm+1,n+1 (z1 , z2 |q ) m+β
(10.43)
(β )
(1 − q )wm−1,n−1 (z1 , z2 |q ). n
( α,β )
11. The polynomials {Mn
( z1 , z2 |q )}
For α , γ > −1, the little q-Jacobi polynomials k n ( q; q )n (qα+γ +n+1 ; q )n−k q(2 ) (−x )n−k n (q; q )k (q, qα+1 ; q )n−k q (2 )
k=0
(11.1) satisfy the orthogonality relation (6.1) with
d μ ( x; γ |q ) =
∞ (qx; q )∞ x δ ( x − qk ), (qγ +1 x; q )∞ k=0
(q, qα+γ +n+1 ; q )∞ (q; q )n qn(α+1) , α +1 ; q ) (1 − qα +γ +2n+1 ) ∞ (q n
(qα+1 , qγ +n+1 ; q )
(11.2) [38]. Following our general construction in Section 6, we define the polynomials
(β ,γ )
Mm,n (z1 , z2 |q ) =
z1m−n pn (z1 z2 , qβ +m−n , qγ |q ), m ≥ n, (β ,γ )
Mn,m (z2 , z1 |q ),
m < n.
(11.3)
(β ,γ )
Mm,n (z1 , z2 |q )
(β ) (1 − qβ +m−n )wm,n ( z , z |q ) +1 1 2 = , qβ +m−n−1 (1 − q )
z2m−n w
(10.42)
(β )
n
In other words
w(z1 z2 ; β|q )
(β ) )wm ( z , z |q ) +1,n 1 2
and
n
(β )
w(z1 z2 ; β|q )wm,n (z1 , z2 |q ) =
)wm,n (z1 , z2 |q )
m+β +1
ζn (α , γ |q ) =
and
(β )
(β )
pn ( x; qα , qγ |q ) =
(10.34)
m−n+β +1
+q (10.33)
(β )
= wm,n (z1 , z2 |q ) − wm+1,n+1 (z1 , z2 |q ), (10.41)
Let
xα .
(10.40)
we can derive the recurrences in the following theorem.
qβ +m−1 θ 2
w(x; α ) = (qx; q )∞
2 ( q; q )n q (−1 )n−k (q; q )k (q, qα+1 ; q )n−k
z2 wm+1,n (z1 , z2 |q )
where θq,z = zDq,z . This gives the following theorem.
wm,n (z1 , z2 |q )
(q; q ) q(α+1)n (q; q )n ζn (α ) = α+1∞ . (q ; q )∞ (qα+1 ; q )n
2 qn θq,z f− 1
(10.38)
(10.37)
(z1 z2 ; β|q )
=
k n (qβ +γ +1 ; q )m+n−k q(2 ) z1m−k (−z2 )n−k (q; q )n (qβ +1 ; q )m−n , n (q; q )k (q; q )n−k (qβ +1 ; q )m−k (qβ +γ +1 ; q )m q(2 ) k=0
(11.4) (β +1,γ )
z1 Mm,n
,γ ) (z1 , z2 |q ) = Mm(β+1 ( z , z |q ) ,n 1 2
(11.5)
for m ≥ n. Now Theorem 6.1 leads to:
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Theorem 11.1. For m, n, s, t = 0, 1, . . . and β , γ > −1 we have the following orthogonality relation
(β ,γ )
R2
(β ,γ )
Mm,n (z, z¯|q )Ms,t
(z, z¯|q )r2β dμ(r2 ; γ |q )dθ
(11.6)
= ζm∧n (|m − n| + β , γ |q )δm,s δn,t , where dμ(x; γ |q), ζ n (α , γ |q) are given in (11.2).
ck (n, α ) =
( q; q )n
( q; q )k
(q, qα+1 ; q )
(−1 ) . (n2 )−(2k ) n−k q n−k
n−k
(β ,γ )
(β ,γ )
q−α − 1 − q1−n−α x − qγ +2 (2 − qn )x 2 (1 − qγ +2 x )θq,x y (x ) + θq,x y(x ) 1−q 1 + q1−α x − (q1−n−α + qγ +n+2 )x − y ( x ) = 0. ( 1 − q )2
qn−m−β − 1 − q1−m−β z1 z2 − qγ +2 (2 − qn )z1 z2 (1 − q )(1 − qγ +2 z1 z2 )
θq,z2 f
1 + (qn+1−m−β − q1−m−β − qγ +n+2 )z1 z2 − f =0 (1 − q )2 (1 − qγ +2 z1 z2 )
+
f+
θq,z1 f
(1 − q )2 (1 − qγ +2 z1 z2 )
f = 0.
When β , γ > −1 the raising and powering operators for the polynomials {pn (x; qα , qγ |q)} with
(qx; q )∞ xα . (qγ +1 x; q )∞
(11.13)
lead to the following ladder operators (β ,γ )
Mm,n (z1 , z2 |q ) − Mm,n (z1 , qz2 |q ) =−
q1−n (1 − qn )(1 − qm+β +γ −1 ) (β ,γ +1 ) z2 Mm,n−1 (z1 , z2 |q ), 1 − qm−n+β
where m ≥ n, and
(β ,γ )
Dq−1 ,z1 w(z1 z2 ; qβ , qγ |q )Mm,n (z1 , z2 |q )
f ( z1 , z2 ) =
(12.1)
∞
a j,k z1j z2k
(12.2)
j,k=0
if and only if
f ( z1 , z2 ) =
∞
a j,o
j=1
+
n
n! z1j (β + j ) L ( z1 z2 ) ( β + j + 1 )n n
a0,k z2k 2 F1
−n + k, 1 z z . k + 1, β + 1 1 2
(12.3)
Proof. Substitute the power series (12.2) for f in (8.25) and equate coefficients of like powers of z1 and z2 to find that
a j,k =
k−1−n a , k(β + j ) j−1,k−1
j, k > 0.
We iterate this and find that
⎧ ⎪ ⎨
(−n )k a j−k,0 , j≥k k! ( β + j − k + 1 )k = ⎪ ⎩ (k − j − n ) j (k − j )! a0,k− j , j ≤ k, k! ( β + 1 ) j
(11.12)
(β ,γ )
∂∂ z2 f = −n f,
Theorem 12.2. In order for the equation
q1−n − q1+m−n+β + qγ +2 (qm+β + q2(m−n+β ) − 1 ) z1 z2 − q2(m−n+β )
w ( x; qα , qγ |q ) =
z1
This proves the theorem.
(1 − q )(1 − qγ +2 z1 z2 )
has a power series solution
a j,k (11.11)
β − z1 z2
k=0
Theorem 11.2. For m ≥ n and β , γ > −1, the function f = ( β ,γ ) Mm,n (z1 , z2 |q ) satisfies
z1 z2 2qγ +2 − q1−n − qm+β +γ +2 + qm−n+β − 1
∂∂ z1 ∂∂ z2 f +
(11.10)
where y(x ) = pn (x; qα , qγ |q ), θq,z = zDq,z .
θ
where m ≥ n + 1.
Theorem 12.1. The partial differential equation (8.25), namely,
that the polynomials {pn (x; qα , qγ |q)} satisfy the following second order difference equation
2 q,z1
z1 (1 − qm−n+β )w(z1 z2 ; qβ , qγ −1 |q ) (β ,γ −1) Mm,n+1 (z1 , z2 |q ), (11.16) z2 qm−n+β −1 (1 − q )
(11.8)
(1 − qβ +m+1 )(1 − qβ +γ +m+1 ) (β ,γ ) = M ( z1 , z2 |q ) (1 − qβ +m−n+1 )(1 − qβ +γ +m+n+1 ) m+1,n qm−n+β +1 (1 − qn )(1 − qγ +n ) (β ,γ ) + M (z1 , z2 |q ). (11.9) (1 − qβ +m−n+1 )(1 − qβ +γ +m+n+1 ) m,n−1 The above formulas hold for β , γ > −1 and m ≥ n. Next we note
and
In this section we study polynomial solutions to partial differential equations. We are looking for polynomial solutions to the second order partial differential equation (8.25). Let f =
∞ j k m,n=0 z1 z2 and substitute in (8.25).
(β ,γ )
2 θq,z f+ 2
(β ,γ )
12. Polynomial solutions to differential equations
z1 Mm,n (z1 , z2 |q )
(11.15)
(11.7)
(1 − qβ +γ +m+n+2 ) (β ,γ ) z2 Mm+1,n (z1 , z2 |q ) qn (1 − qβ +m−n+1 ) = −Mm+1,n+1 (z1 , z2 |q ) + Mm,n (z1 , z2 |q ),
(1 − qm−n+β )w(z1 z2 ; qβ , qγ −1 |q ) (β ,γ −1) Mm,n+1 (z1 , z2 |q ), qm−n+β (1 − q )
z2n−m Dq−1 ,z2 z2m−n w(z1 z2 ; qβ , qγ |q )Mm,n (z1 , z2 |q ) =
Following the construction of §6 we derive the recurrence below via the choice
(qα+γ +n+1 ; q )
=
19
(11.14)
∂z1 ∂z2 f +
β z1
− z2
∂z2 f = λ f
(12.4)
to have a polynomials solution in z2 , it is necessary and sufficient that λ = −n, n = 0, 1, 2, . . . , in which case the function f will be given as in Theorem 12.1. References [1] S.T. Ali, F. Bagarello, G. Honnouvo, Modular structures on trace class operators and applications to Landau levels, J. Phys. A 43 (2010) 17. 105202. [2] W.A. Al-Salam, T.S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976) 16–28. [3] G.E. Andrews, R.A. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. [4] P. Appell, J.K. de Fériet, Fonctions Hypergéométrique et Hypersphérique, Polynomes D’hermite, Gauthier–Villars, Paris, 1926. [5] R.A. Askey, A Note on the History of Series, in: Technical Report 1532, Mathematics Research Center, University of Wisconsin. [6] R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Academic Press, 1964. [7] J.L. Burchnal, A note on the polynomials of Hermite, Q. J. Math. 12 (1941) 9–11.
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[8] L. Carlitz, A set of polynomials in three variables, Houston J. Math. 4 (1978) 11–33. [9] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. Reprinted by Dover, New York, 2003. [10] N. Cotfas, J.P. Gazeau, K. Górska, Complex and real Hermite polynomials and related quantizations, J. Phys. A 43 (2010) 14. 305304. [11] C. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, 2nd ed., Cambridge University Press, Cambridge, 2014. [12] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, 2, McGraw-Hill, New York, 1953. [13] E. Feldheim, La transformation de Gauss á plusieurs variables. Application aux polynomes d’Hermite et á la généralisation de la formule de mehler, Pont. Acad. Sci. Comment. 6 (1942) 1–25. [14] J.C. Ferreira, V.A. Menegatto, Eigenvalues of integral operators defined by smooth positive definite kernels, Integral Equ. Oper. Theory 64 (1) (2009) 61–81. [15] P.G.A. Floris, Addition formulas for q-disk polynomials, Compos. Math. 108 (1997) 123–149. [16] P.G.A. Floris, A noncommutative discrete hypergroup associated with q-disk polynomials, J. Computa. Appl. Math. 68 (1996) 69–78. [17] P.G.A. Floris, H.T. Koelink, Commuting q-analogue of the addition formula for disc polynomials, Constr. Approx. 13 (1997) 511–535. [18] D. Foata, Some hermite polynomial identities and their combinatorics, Adv. Appl. Math. 2 (1981) 250–259. [19] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd ed., Cambridge University Press, Cambridge, 2004. [20] A. Ghanmi, A class of generalized complex Hermite polynomials, J. Math. Anal. Appl. 340 (2008) 1395–1406. [21] A. Ghanmi, Operational formulae for the complex Hermite polynomials h p,q (z, z¯ ), Integral Transform. Spec. Funct. 24 (2013) 884–895. [22] K. Gorska. Private communication. [23] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 2012. [24] A. Intissar, A. Intissar, Spectral properties of the cauchy transform on 2 l 2 (C; e−|z| dz ), J. Math. Anal. Appl. 31 (2006) 400–418. [25] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in one Variable, paperback, Cambridge University Press, Cambridge, 2009. [26] M.E.H. Ismail, Analytic properties of complex Hermite polynomials, Trans. Am. Math. Soc. 368 (2016) 1189–1210. [27] M.E.H. Ismail, P. Simeonov, Complex Hermite polynomials: their combinatorics and integral operators, Proc. Am. Math. Soc. 143 (2015) 1397–1410. [28] M.E.H. Ismail, P. Simeonov, Combinatorics and operator properties of the n-dimensional Hermite polynomials, J. Math. Anal. Appl. (2017). in press. [29] M.E.H. Ismail, J. Zeng, Two variable extensions of the Laguerre and disc polynomials, J. Math. Anal. Appl. 424 (2015a) 289–303. [30] M.E.H. Ismail, J. Zeng, A combinatorial approach to the 2d-Hermite and 2d-Laguerre polynomials, Adv. Appl. Math. 64 (2015b) 70–88. [31] M.E.H. Ismail, R. Zhang, Classes of bivariate orthogonal polynomials, symmetry, integrability and gemoetry: methods and applications (SIGMA) 12 (2016) 37. [32] M.E.H. Ismail, R. Zhang, Kibble-slepian formula and generating functions for 2d polynomials, Adv. Appl. Math. 80 (2016b) 70–92. [33] M.E.H. Ismail, R. Zhang, On some 2d orthogonal q-polynomials, Trans. Am. Math, Soc. (2016) in press. [34] K. Ito, Complex multiple Wiener integral, Jpn. J. Math. 22 (1952) 63–86. [35] S. Karlin, G. Szgeo˝ , On certain determinants whose elements are orthogonal polynomials, J. Anal. Math. 8 (1960/61) 1–157. Reprinted in “Gabor Szego˝ Collected Papers”, vol. 3, R. Askey, ed., Birkhäuser, Boston, 1982, p. 605–761.
[36] D. Karp, Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, in: Analytic Extension Formulas and their Applications, Kluwer Academic Publishers, 2001, pp. 169–188. [37] W.F. Kibble, An extension of theorem of mehler on Hermite polynomials, Proc. Camb. Philos. Soc. 41 (1945) 12–15. [38] R. Koekoek, R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogues, in: Reports of the Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, 1998. No. 98-17. [39] T.H. Koornwinder, The addition theorem for Jacobi polynomials III, in: Completion of the proof, Math. Centrum, Report TW 135/72, Math. Centrum, Amsterdam, 1972. [40] T.H. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: R. Askey (Ed.), Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 435–495. [41] J.D. Louck, Extension of the Kibble–Slepian formula for Hermite polynomials using Boson operator methods, Adv. Appl. Math. 2 (1981) 239–249. [42] C.D. Maldonado, Note on orthogonal polynomials which are “invariant in form” to rotations of axes, J. Math. Phys. 6 (1965) 1935–1938. [43] D.R. Myrick, A generalization of the radial polynomials of F. zernike, J. SIAM Appl. Math. 14 (1966) 476–489. [44] I. Novik, A. Postnikov, B. Sturmfels, Syzygies of oriented matroids, Duke J. Math. 111 (2002) 287–317. [45] E.D. Rainville, Special Functions, Macmillan, New York, 1960. [46] J. Riordan, Inverse relations and combinatorial identities, Am. Math. Mon. 71 (1964) 485–498. [47] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York–London-Sydney, 1968. [48] L. Sa˘ piro, Special functions related to the representation of the group SU(n), of class i with respect to s(n − 1 )(n ≥ 3), in Russian, Izv. Vyss. Uc̎eb. Zaved. Mat. 71 (4) (1968) 97–107. [49] D. Slepian, On the symmetrized Kronecker power of a matrix and extensions of Mehler’s formula for Hermite polynomials, SIAM J. Math. Anal. 3 (1972) 606–616. [50] G. Szego˝ , Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, 1975. [51] K. Thirulogasanthar, G. Honnouvo, A. Krzyzak, Coherent states and Hermite polynomials on quaterionic Hilbert spaces, J. Phys. A 16 (2010) 13. 385205. [52] S.J.L. van Eijndhoven, J.L.H. Meyers, New orthogonality relations for the Hermite polynomials and related Hilbert spaces, J. Math. Anal. Appl. 146 (1990) 89–98. [53] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1927. [54] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge University Press, Cambridge, 1933. [55] A. Wünsche, Laguerre 2d-functions and their applications in quantum optics, J. Phys. A 31 (1998) 8267–8287. [56] A. Wünsche, Transformations of Laguerre 2d-polynomials and their applications to quasiprobabilities, J. Phys. A 21 (1999) 3179–3199. [57] A. Wünsche, Generalized zernike or disc polynomials, J. Comp. App. Math. 174 (2005) 135–163. [58] F. Zernike, Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode, Physica 1 (1934) 689–701. [59] F. Zernike, H.C. Brinkman, Hypersphdrische funktionen und die in spharischen bereischen orthogonalen polynome, Nederl. Akad. Wetensch. Indag. Math. 38 (1935) 161–171.
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Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–20
Contents lists available at ScienceDirect
Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems
Review Paper
A review of multivariate orthogonal polynomialsR Mourad E.H. Ismail a, Ruiming Zhang b,∗ a b
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA College of Science, Northwest A&F University Yangling, Shaanxi 712100, PR China
a r t i c l e
i n f o
Article history: Received 24 September 2016 Accepted 23 November 2016 Available online xxx MSC: 33C50 33D50 33C45 33D45
a b s t r a c t This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user. © 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Keywords: Disc polynomials Zernike polynomials 2D-Laguerre polynomials q-2D-Laguerre polynomials Generating functions Ladder operators q-Sturm–Liouville equations q-integrals q-Zernike polynomials 2D-Jacobi polynomials q-2D-Jacobi polynomials Connection relations Biorthogonal functions Generating functions Rodrigues formulas Zeros
1. Introduction This is a survey paper of certain classes of orthogonal polynomials in two complex variables and the multivariate Hermite polynomials. The multivariate real Hermite polynomials go back to Hermite and they were studied in some detail in the book by Appell and Kampé de Feriet [4]. We felt it will be beneficial to collect scattered old and new results on 2D and nD polynomials in one place. This paper complements and updates the influential survey
R Part of this work was completed at BIRS in Banff and the authors acknowledge the support from BIRS. Research of R. Zang was partially supported by Chinese national natural science foundation, Grant no. 11371294. ∗ Corresponding author. E-mail address: [email protected] (R. Zhang).
article [40] by Koorwinder which is now 40 years old but is still a valuable reference on the subject. One of the earliest orthogonal polynomials are the Legendre polynomials which are orthogonal with respect to dx = (Lebesque measure) on the unit interval [−1, 1]. They are the spherical harmonics in R2 . The ultraspherical (or Gegenbauer) polynomials are the spherical harmonics on Rm and are orthogonal with respect ν +3/2 ) to √( (1 − x2 )ν dx on the unit interval [−1, 1]. If we replace x π (ν +1 ) √ 2 by x/ ν and let ν → ∞ the measure becomes π −1/2 e−x dx and the ultraspheical polynomials, properly scaled, tend to Hermite polynomials. The next sequence of polynomials in this hierarchy are the Jacobi polynomials which are orthogonal on the unite interval with respect to
2−α −β −1 (α +β +2 ) α (α +1 )(β +1 ) (1 − x ) (1 +
x )β dx. The Laguerre poly-
nomials arise when x is replaced by −1 + 2x/α and α → ∞. It is worth mentioning that none of the one variable polynomials is
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named after the person who first discovered them, see the interesting paper by Askey [5]. The 2D history is somewhat parallel. In the late 1920s Frits Zernike (Nobel prize for physics in 1953) was working on optical problems involving telescopes and microscopes. He introduced polynomials orthogonal on |z| ≤ 1 with respect to the area measure, so this is the 2D analogue of Legendre polynomials, see [58] and [59]. The more general disc or Zernike polynomials are orthogonal with respect to (1 − x2 − y2 )ν dxdy on the unit disc. Again √ √ if we replace (x, y) by (x/ ν , y/ ν ) and let ν → ∞ this measure, 2 2 properly normalized becomes e−x −y dxdy on R2 . The polynomials become the 2D-Hermite polynomials introduced by Ito in [34] in a different way. They are defined by
Hm,n (z1 , z2 ) =
m ∧n k=0
m k
n (−1 )k k!z1m−k z2n−k , k
f (z ) − f (qz ) (Dq,z f )(z ) = , ( 1 − q )z 1 − qα [α ]q = , 1−q
(δq,z f )(z ) = z(Dq,z f )(z ),
m ∨ n = max{m, n},
Aq (z ) :=
∞ (−z )n n2 q , ( q; q )n
(1.5)
n=0
ϑ4 (z; q ) :=
∞
(−1 )n qn zn . 2
(1.6)
n=−∞
Note that our ϑ4 is slightly different from ϑ4 in Whittaker and Watson [53]. The q-Laguerre polynomials are [25, Section 21.8] (α )
Ln
n (qα+1 ; q )n n qk ( k+α ) (z ) = (−z )k . ( q; q )n k (qα +1 ; q )k q
(1.7)
k=0
2. The complex Hermite polynomials
(1.2)
m ∧ n = min{m, n} (1.3)
and
∂ , δz = z∂z , δq,z = zDq,z . ∂z
For completeness we recall the definition of the Ramanujan function Aq (z) and ϑ4 ,
(1.1)
where a∧b := min {a, b}. In 1966 Myrick [43] considered only the radial part of the polynomials orthogonal with respect to the Jacobi type measure (x2 + y2 )α (1 − x2 − y2 )β dxdy on the unit disc. An addition theorem for the disc polynomials were found by Sa˘ piro [48] and Koornwinder [39]. Maldonado [42] very briefly considered a class of radial functions which can be used to generate 2D orthogonal polynomials. Floris [15] introduced a q-analogue of the disc polynomials proved an addition theorem for them, see also Floris and Koelink [17]. Wünche [57] also studied the disc polynomials. We shall use the notations
∂z =
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The complex Hermite polynomials are the usual Hermite polynomials {Hn (x + iy )}n≥0 , x, y ∈ R when viewed as functions of the two variables x and y. Many of the algebraic properties of Hermite polynomials {Hn (x)}n ≥ 0 hold when x is taken as a complex variable. In particular we have the generating function [25,45,50] ∞ Hn (z ) n t = exp(2zt − t 2 ). n!
(2.1)
n=0
Theorem 2.1. Assume that
1 1 =1+ . a b
0 < a < b,
(2.2)
Then the complex Hermite polynomials satisfy the orthogonality relation, [36,52],
R2
Hm (x + iy )Hn (x − iy ) e
−ax2 −by2
π
a+b dx dy = √ 2 n! ab ab n
δm,n (2.3)
(1.4)
In addition we shall follow the standard notation for shifted factorials, hypergeometric functions and their q-analogues as in [3,19], and [25]. In Section 2 we treat the Complex Hermite polynomials {Hn (x + iy )}n≥0 . They are orthogonal on R2 and are polynomials in the two real variables x and y with complex coefficients. We state their orthogonality relations and show that they are eigenfunctions of a 2D-Laplace transform. In Section 3 we treat Ito’s 2D-Hermite polynomials in some detail. This includes orthogonality, differential recurrence relations, pure recurrence relations as well as linear and multilinear generating functions. Many of the multilinear generating functions are kernels of integral operators with explicit eigenvalues and eigenfunctions. This is typical of the material treating specific sets of orthogonal polynomials in the subsequent sections. This section is followed by Section 4 which is a systematic study of their q-analogues introduced by the present authors in [33]. Section 5 contains a survey of the n-dimensional Hermite polynomials introduced by C. Hermite in 1860s. They form a biorthogonal system of polynomials. Section 5 contains recent results from [28]. In particular we mention combinatorial interpretations of the integrals of products of the polynomials times their weight functions. In Section 6 we briefly describe a class of polynomials we introduced in [31]. This contains a general construction of bivariate orthogonal polynomials from a univariate system of polynomials orthogonal on a half line. Sections 7–11 contains important examples of this construction. In each section a specific system of orthogonal polynomials is introduced and its important properties are recorded. The paper concludes with Sections 12 where we discuss polynomial solutions to certain second order partial differential equations.
n
as well as the orthogonality relations [22]
Hm (x + iy )Hn (x + iy )e−ax −by dx dy π 2 2 = Hm (x − iy )Hn (x − iy )e−ax −by dx dy= √ 2n n! δm,n . (2.4) R2 ab 2
2
R2
The proof follows from the generating function (2.1). 2 2 The weight function e−ax −by , on R has the moments
R2
R2
(x ± iy )2n+1 e−ax (x ± iy )2n e−ax
2
2
−by2
−by2
d x d y = 0,
π
d x d y = √ ( 1/2 )n , ab
(2.5)
provided that the condition (2.2) is satisfied. Ismail and Simeonov [27] studied the combinatorics of the complex Hermite polynomials. One of their combinatorial results established the more general orthogonality relation
R2
Hm (α x + β y )Hn (γ x + δ y )e−ax
π
= √ 2n n! ab
αγ a
+
βδ
2
−by2
dx dy
n
b
δm,n
(2.6)
where
a > 0,
b > 0,
α2 a
+
β2 b
=
γ2 a
+
δ2 b
= 1.
(2.7)
Formula (2.6) implies that for
√ α/ a √ δ/ a
√ β/ b √ ∈ SO(2, R ), δ/ b
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then Hn (α x + β y ) is orthogonal to Hm (δ x + γ y ) unless m = n = 0. It is also clear that the orthogonality relations (2.3) and (2.4) are special cases of (2.6). Theorem 2.2. Suppose that we have two colored multisets. The first, of color I, has size m = (m1 , m2 , . . . , mk ) ∈ Nk0 and the second, of color II, has size n = (n1 , n2 , . . . , nk ) ∈ Nk0 . We match elements from different sets, and to each pair of matched elements we assign weight 1 if the elements have the same color and weight 1/a + 1/b if the elements have different colors. Then the total weight of all perfect matchings of this type is the number B(m, n) defined by
B ( m, n ) =
√ ab
π
2 − 2 ( | m| +| n| ) 1
k R2
× e−ax
2
−by2
euz1 +vz2 =euv
m,n≥0
then,
z1j z2k =
m, n, ≥ 0, m + = j, n + = k
Hn (z ),
(2.9)
√ ab 2 2 2 2 eiz(ξ −iη )+(z +(ξ −iη ) )/2 Hn (ξ − iη ) e−aξ −bη dξ dη √ π 2 R2 = in Hn (z ),
(2.10)
n ∈ N0 , where a and b satisfy conditions (2.2). This theorem indicates that the complex Hermite polynomials are eigenfunctions of a 2-dimensional Fourier transform. However, ∞ we could not use the orthogonal system S = {Hn (x + iy )}n=1 ∪ {1} ∪ ∞ {Hn (x − iy )}n=1 to define a 2d Fourier transform. First we observe that Hn (x − iy ) is not an eigenfunction. Furthermore, the system S √ 2 2 is not complete. It has the same L2 (R2 , πab e−ax −by dxdy ) closure ∞ ∞ as {(x + iy )n }n=1 ∪ {1} ∪ {(x − iy )n }n=1 , which does not contain all ∞ n m 2 2 of {x y }n,m=0 . For example, x , y is not in the closure. Moreover, in order to do Fourier analysis we also need both 2.3 and 2.4 under the condition 2.2 to ensure the orthogonality of system S. To require they are eigenfunctions correspond eigenvalue ±i, ±1, from Theorem 2.3 we also need to have the condition 1a + 1b = 1. But the two conditions imply that a = 1, b = ∞, which is impossible.
k n
j+k−m−n !Hm,n (z1 , z2 ) 2
j ≥ m ≥ 0, k ≥ n ≥ 0
n
j m
j ≥ m ≥ 0, k ≥ n ≥ 0 ×
j!k! Hm,n (z1 , z2 ), m!n! !
=
√ ab 2 2 2 2 eiz(ξ −iη )+(z +(ξ −iη ) )/2 Hn (ξ + iη ) e−aξ −bη dξ dη √ π 2 R2 b+a i ab
∂ j+k euz1 +vz2 j ∂ z1 ∂ z2k z1 =z2 =0
(2.8)
Theorem 2.3 [27]. The following integral relations hold
=
m,n, ≥0
=
Observe that the orthogonality relation (2.3) is the special case k = 1 of Theorem 2.2.
um vn um+ vn+ Hm,n (z1 , z2 )= Hm,n (z1 , z2 ), m!n! m!n! !
z1j z2k =
j=1
dx dy.
By 3.1 we also get
which gives
[Hm j (x + iy )Hn j (x − iy )]
3
=
j ≥ m ≥ 0, k ≥ n ≥ 0
j m
k ( j − m )!Hm,n (z1 , z2 ) n
j m
k (k − n )!Hm,n (z1 , z2 ). n
The combinatorics of these polynomials have been studied in the recent work [27]. We note that Carlitz rediscovered these polynomials in the more recent work and found their generating functions [8]. It is important to think of the 2D-Hermite polynomials as polynomials in two complex variables or four real variables but their orthogonality is on the complex plane or on R2 . From the generating function it is easy to see that the polynomials satisfy the three term recurrence relations
z1 H p,q (z1 , z2 ) = qH p,q−1 (z1 , z2 ) + H p+1,q (z1 , z2 ) z2 H p,q (z1 , z2 ) = pH p−1,q (z1 , z2 ) + H p,q+1 (z1 , z2 ).
(3.3)
They also satisfy the connection relation
m + n
Hm,n (z, z¯ ) = m!n!
i 2
m n i j+k (−1 ) j+n j!k! j=0 k=0
H j+k (x )Hm+n− j−k (y ) × , ( m − j )! ( n − k )!
(3.4)
and its inverse 3. Ito’s 2D-Hermite polynomials The complex Hermite polynomials {Hm, n (z1 , z2 )} defined by (1.1) were introduced by Ito [34] in his study of complex multiple Wiener integral and were used in [1] to study Landau levels. They were applied in [51] to coherent states, and in [55,56] to quantum optics and quasi probabilities, respectively. See also [10,20,21]. They are essentially the same as the polynomials in [11, (2.6.6)]. They also appeared in combinatorics in the work of Novik et al. [44]. Their exponential generating function is ∞ m, n=0
Hm,n (z1 , z2 )
1
π
R2
Hm,n (x + iy, x − iy )H p,q (x + iy, x − iy ) e−x
j,m≥0
r! s! is (−1 )m− j Hm,r+s−m (z, z¯ ). j ! ( r − j )! ( m − j )! ( s + j − m )! (3.5)
Also note the symmetry relation
Hm,n (z, z¯ ) = Hn,m (z, z¯ ).
(3.6)
The linearization of products is given by
Hm1 ,n1 (z1 , z2 )Hm2 ,n2 (z1 , z2 )=
n1 !n2 !Hm +m − j−k,n +n − j−k (z1 ,z2 ) 1 2 1 2 j,k j!k!(m1 − j )!(n1 −k )!(m2 −k )!(n2 − j )! ,
(3.7)
um vn = euz1 +vz2 −uv , m! n!
(3.1)
2
−y2
dx dy
= m! n! δm,p δn,q .
and its inverse is, [21],
H p+m,q+n (z1 , z2 ) p! q! m! n!
and they satisfy the orthogonality relation, [24] and [20]
Hr (x )Hs (y ) =
= (3.2)
p∧n q∧m (−1 ) j+k H p− j,q−k (z1 , z2 ) Hm−k,n− j (z1 , z2 ) . j!k! ( p − j )!(q − k )! (m − k )!(n − j )!
(3.8)
j=0 k=0
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When m = n + s ≥ n the polynomials Hm,n (z1 , z2 ) are related to Laguerre polynomials via
Theorem 3.5, finally prove Theorem 3.2 by analytic continuation. It is interesting to see this can be carried out for a larger domain.
Hn+s,n (z1 , z2 ) = (−1 )n n! z1s Ln(s ) (z1 z2 ).
Corollary 3.3. For N ∈ N, let W = ρ1 eiθ1 , . . . , ρN eiθN that ρ j > 0, θ j ∈ R for j = 1, . . . , N in (3.15), H, IN K, cj and rj are the same as in Theorem 3.1 but H is not necessarily Hermitian, then
(3.9)
The symmetry relation (3.6) shows that Hn,n (z, z¯ ) is real. The Rodrigues formulas are
Hm,n (z1 , z2 ) = (−1 )m+n ez1 z2 ∂zm2 ∂zn1 e−z1 z2 , Hm,n (z1 , z2 ) = (−1 )m+n ez1 z2 ∂zn1 ∂zm2 e−z1 z2 , where ∂z := ∂∂ . We note that
∂z
1 = 2
z
∂ ∂ −i , ∂x ∂y
∂z¯
1 = 2
(3.10)
det (IN + H )−1 exp W ∗ H (IN + H )−1W k jk c j
= K Nj=1 Nk=1 −h j,k k ,...,k 1, j
∂ ∂ +i . ∂x ∂y
n m
Hm,n (z1 , z2 ) = (−1 )n ez1 z2 ∂z1 z1 e−z1 z2 .
(3.11)
(3.12)
N, j
ρ j e iθ j
(−∂z1 + z2 ) f (z1 , z2 ) (3.13)
k=0
(3.14)
The Kibble–Slepian formula is a multilinear generating function for Hermite polynomials and is due to Kibble [37] and later was proved using Fourier transforms by Slepian [49]. Foata [18] gave a nice combinatorial proof while Louck [41] used Boson operator calculus to give a new proof of this remarkable result. The norm of a matrix A = {a j,k : 1 ≤ j, k ≤ N} is the Frobenius or the Hilbert–
N 2 Schmidt norm j,k=1 |a j,k | , [23]. Ismail [26] proved the follow-
0
u . 0
v
∞ n1,2 ,n2,1 =0
u n1,2 vn2,1 Hn ,n (z1 , z1 )Hn2,1 ,n1,2 (z2 , z2 ) n 1,2 ! n 2,1 ! 1 ,2 2 ,1
= (1 − uv )−1 exp
a
u , b
H=
Det[(IN + H )−1 ] exp ZH (IN + H )−1 Z ∗ n
(h ) j,k = K 1≤ j,k≤N nj,k ! Hr1 ,c1 (z1 , z1 ), . . . , HrN ,cN (zN , zN ),
This leads to the bilinear generating function
(3.15)
where K = (n j,k : 1 ≤ j, k ≤ N ) is a general matrix with nonnegative integer entries, ck is the sum of the elements of K in column k and rj is the sum of the elements of K in row j, that is
n j,k ,
j=1
rj =
n j,k .
Theorem 3.2. Following the notation in Theorem 3.1 we let δ be a fixed real number such that 0 < δ < 31N , then (3.15) is also true for
1 ≤ j, k ≤ N
a, b ∈ R
an1,1 bn2,2 un1,2 vn2,1 Hn +n ,n +n (z1 , z1 ) n 1,1 ! n 2,2 ! n 1,2 ! n 2,1 ! 1 ,1 1 ,2 1 ,1 2 ,1 × Hn2,1 +n2,2 ,n1,2 +n2,2 (z2 , z2 )
= [(1 + a )(1 + b) − uv]−1 × exp
(3.16)
k=1
It is important to note that the multilinear kernels given by Theorem 3.1 will always be positive. The following results, namely Theorem 3.2 and Corollary 3.3, are due to Ismail and Zhang [32].
h j,k ≤ δ,
v
n j,k ≥0,1≤ j,k≤2
j,k
ck =
(3.19)
Example 2 (General2 × 2 Case). Ismail [26] considered the general 2 × 2 case
Theorem 3.1. Let Z = (z1 , z2 , . . . , zN ), and H be an N × N Hermitian matrix with H < 1, and IN is an N × N identity matrix. Then
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 . 1 − uv
The Poisson kernel (3.19) is stated in [56] without proof. Carlitz discovered the 2D-Hermite polynomials independtly of Ito’s work and established (3.19). He did not prove the orthogonality relation.
ing
N
j
In this case Theorem 3.1 gives the Poisson kernel. Indeed in the sum in (3.15) we must have n1,1 = n2,2 = 0 and we get
is from [21] and extends a one variable result due to Burchnall [7]. Another operational formula is
N
cj
Example 1 (Poisson Kernel). As an example consider the case N = 2 with
H=
r j −c j (r j −c j ) 2 L ρ ,
where {Ln(α ) (x )} are Laguerre polynomials.
n
Hm,n (z1 , z2 ) = exp(−∂z1 ∂z2 )(z1m z2n ).
This follows from Theorem 3.2 and (3.9). It must be noted that Ismail and Zeng [30] gave a purely combinatorial proof of (3.15) but as a formal power series.
The operational formula n (−1 )k Hm,n−k (z1 , z2 ) k −m = n! ∂z1 z 1 f ( z 1 , z 2 ) , k! ( n − k )!
(3.18)
Note that (3.10) is equivalent to the formulas
Hm,n (z1 , z2 ) = (−1 )m ez1 z2 ∂zm2 z2n e−z1 z2 ,
T
(3.20)
(a(1 + b) −uv )z1 z1 + uz1 z2 + vz1 z2 + (−uv + b(1 + a ))z2 z2 . (1 + a )(1 + b) − uv
An important observation is that (3.19) is essentially a generating function for Laguerre polynomials. Write the series in (3.19) as a sum over n1,2 ≥ n2,1 plus a sum over n1,2 ≤ n2,1 minus the sum over n1,2 = n2,1 . The result is ∞
n!
( n + s )! s,n=0
(3.17)
+
It is known that a multivariate analytic function f (z1 , . . . , zn ) at (0, . . . , 0 ) can be expanded in a convergent polynomial series, [6]. Then one may prove Theorem 3.2 on a domain, possibly all of ||H|| < 1, in the following steps: first prove the polynomial series is exactly the right hand side of 3.15, then prove both sides of 3.15 are analytic inside the domain by applying the estimate in
−
and H is not necessarily Hermitian.
∞
un+s vn (z1 z¯2 )n Ln(s ) (|z1 |2 )Ln(s ) (|z2 |2 )
n!
( n + s )! s,n=0 ∞
un vn+s (z¯1 z2 )n Ln(s ) (|z1 |2 )Ln(s ) (|z2 |2 )
( uv )n Ln ( |z1 |2 )Ln ( |z2 |2 )
n=0
= (1 − uv )−1 exp
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 . 1 − uv (3.21)
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The third series can be summed by the case α = 0 of Hille–Hardy formula, [25, (4.7.20)] ∞ n=0
n! r n L(α ) (x )Ln(α ) (y ) ( α + 1 )n n
= (1 − r )
−α −1
r (x + y ) exp − 1−r
0 F1
(−; α + 1; xyr (1 − r )−2 ). (3.22)
It will be of interest to give a direct proof of (3.21) using properties of Laguerre polynomials. We call a sequence of polynomials {pm, n (z1 , z2 )} a bivariate Appell sequence if its exponential generating function is of the type ∞ m,n=0
um vn pm,n (z1 , z2 ) = A(u, v ) exp(uz1 + vz2 ), m!n!
(3.23)
where A(u, v ) is analytic in u and v and we normalize our formulas by A(0, 0 ) = 1. Theorem 3.4 [26]. The only bivariate Appell sequence pn (z1 , z2 ) such that pn (z, z¯ ) is orthogonal on R2 is {Hm,n (z, z¯ )}. For properties of Appell polynomials in one variable we refer the interested reader to [45]. Theorem 3.5 [32]. For all m, n ∈ Z+ and z ∈ C we have
|Hm,n (z, z )| ≤ e|z|
2
√ m! · n!.
Theorem 3.5 is very useful in justifying interchanging sums, or sums and integrals. We now construct some integral operators. Let
K (z1 .z¯1 , z2 , z¯2 ; u, v ) =
1
π
(1 − uv )−1 exp
−uvz1 z1 + uz1 z2 + vz1 z2 − uvz2 z2 , 1 − uv (3.24)
of (3.26) can be expanded in power series in u and v when |u| < 1 and |v| < 1. We next consider linearization of products of 2D-Hermite polynomials. Note that (3.7) can be written in the form
Hm1 ,n1 (z, z¯ )Hm2 ,n2 (z, z¯ )
n1 !Hm1 +m2 −s,n1 +n2 −s (z,z¯ ) = s m ! s! (n −s )!(m −s )! 3 F2 1
(T f )(z1 , z¯1 ) =
R2
×e
K (z1 , z¯1 , x + iy, x − iy, u, v ) f (x + iy, x − iy ) −x2 −y2
dxdy,
(3.25)
2 2 (e−x −y dxdy, R2 ).
defined on L2 We know that the polynomials 2 2 {Hr (x)Hs (y)}r, s ≥ 0 are dense in L2 (e−x −y dxdy, R2 ). Hence formula (3.5) shows that {Hm,n (z, z¯ )}m,n≥0 are also dense in the same space, with z = x + iy. Therefore the polynomial system {Hm,n (z, z¯ )}m,n≥0 −x2 −y2
is complete in L2 (e dxdy, R2 ). Then the orthogonality relation (3.2) and the symmetry (3.6) imply
R2
K (z1 , z¯1 , x + iy, x − iy, u, v )Hm,n (x + iy, x − iy )e−x
2
−y2
1
It is clear that (3.27) and the orthogonality relation (3.2) give the value of the integral
k R2
Hm j ,n j (z, z¯ )e−x
2
−y2
dxdy,
(3.28)
j=1
when k = 3. This raises the question of the evaluation of a product of k polynomials. For k > 3 this integral does not have a nice closed form but has an interesting combinatorial interpretation; see [27]. Formulas (3.7) and (3.8) are inverse relations in a combinatorial sense, see Riordan [46,47]. Another result which immediately follows from the generating function is the addition formula
Hm,n (z + w, z¯ + w¯ ) m! n! √ √ √ √ p q H j,k ( 2z, 2z¯ )) H p− j,q−k ( 2w, 2w¯ )) −( p+q )/2 =2 . j! k! ( p − j )! ( q − k )! j=0 k=0
(3.29) The above formula was proved in [21] but also follows from the exponential generating functions [26]. In [26] Ismail derived analogues of ∞
Hn+k (x )
n=0
Hn (cx ) =
tn 2 = e2xt −t Hk (x − t ), n!
n/2 k=0
n!cn−2k (1 − c2 )k Hn−2k (x ), k ! ( n − 2 k )!
[25, (4.6.29)] and [25, (4.6.33)]. His results are ∞
um vn = (−1 ) j+k euz+vz¯−uv H j,k (z − v, z¯ − v¯ ), m!n!
Hm+ j,n+k (z, z¯ )
m,n=0
(3.30) and
Hm,n (cz, c¯z¯ ) =
m ∧n
Hm− j,n− j (z, z¯ )
j=0
dxdy
= um vn Hm,n (z1 , z¯1 ).
2
−s, −m1 , −n2 −1 . n1 − s + 1, m2 − s + 1 (3.27)
and consider the integral operator
5
m! n!cm− j c¯n− j (cc¯ − 1 ) j . j! (m − j )! (n − j )! (3.31)
(3.26)
This shows that {Hm,n (z1 , z¯1 )} are the eigenfunctions of T with eigenvalues um vn and the completeness of the system implies that there are no other eigenfunctions. For 0 < u, v < 1 we observe that T is self-adjoint, compact and positive definite. But the series expansion for K cannot be obtained by applying the spectral theorem directly. The expansion obtained from spectral theorem is a re-ordering of {um vn }∞ so that the obtained 1d sequence dem,n=0 creases to 0. It is not hard to see the expansion also provides an example for the generalized Mercer’s theorem, [14]. The Fourier transform in two dimensions corresponds to the special value u = v = i in (3.26). This case was stated explicitly in [55] and can be proved by imitating the classical proof for the one variable Hermite polynomial case in [54]. One cannot however simply put u = v = i in (3.26) because the Poisson kernel formula (3.19) is valid for max{|u|, |v|} < 1, since the right hand side
Notice that by setting z = 1, formula 3.31 gives yet another representation of Hm,n (c, c¯ ) in terms of cm− j c¯n− j (cc¯ − 1 ) j . He also extended
m1 +m2 +···+mr =n
provided that
r j=1
m j =m,
= Hm,n when
r
j=1
m j!
r
j=1
j=1
j=1
j=1
a jz j,
(3.32)
j=1
m
Hm j ,n j (z j , z¯j )
n j =n j=1
r
Hm j (x j )
r 1 = Hn a jx j , n!
a2j = 1, [12, (10.13.40)], to
r
r
mj
r a j
r
a j j a¯j n j m j! n j!
a¯j z¯j ,
(3.33)
j=1
|a j |2 = 1.
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The Hermite polynomials have the generating function [45, Section 106] ∞ ( c )n n t Hn (x ) = (1 − 2xt )−c 2 F0 n! n=0
c/2, (c + 1 )/2 −4t 2 (1 − 2xt )2 . −
Then N is given by
1 N = N!
(3.34) In general (3.34) holds as a formal power series since the 2 F0 diverges, unless c is a negative integer, in which case (3.34) holds as equality between functions. Rainville [45] attributes (3.34) to Brafman for general c and to Truesdell when c = 1. We now extend (3.34) to the complex Hermite polynomials and again our result will be a formal power series identity unless a or b is a negative integer when it becomes an equality. The generating function is
∞
( a )m ( b )n m n u v H
m,n (z, z¯ ) m,n=0 m! n! a, b −a −b = (1 − uz ) (1 − vz¯ ) 2 F0
uv . − (1 − uz )(1 − vz¯ )
(3.35)
We now give a moment integral representation of the Hm, n ’s. Theorem 3.6. Let z = x + iy with x, y ∈ R. The complex Hermite polynomials have the moment representation
Hm,n (iz, iz¯ ) =
im+ n
π
R2
(r + is )m (r − is )n
× exp −(r − x )2 − (s − y )2 drds.
R2
w wn exp {−ww+iz1 w+iz2 w}drds. m
(3.37) Note that Theorems 3.6 and 3.7 are the 2D analogues of
Hn (ix ) 1 = √ ( 2i )n π
e−(y−x ) yn dy, 2
(3.38)
R
e−x R2
2
−y
2
Hm,n (cz, c¯z¯ )dxdy =
R2
(cc¯ − 1 )n n!
One more integral from [26] is
if m = n,
0
e
−x2 −y2
Hm,n (ζ , ζ¯ ) dxdy =
π if m = n.
(3.39)
0 if n − m is odd, C (m, s ) if n − m = 2s,
(3.40)
(3.41)
Several authors considered sign regularity of determinants whose entries are functions. This includes Wronskians and Turánians. One classic paper in the subject is the monumental work of Karlin and Szego˝ [35]. Theorem 3.6 is the key to proving Theorem 3.8 below. Theorem 3.8 [26]. Set z = x + iy and let N be the determinant whose elements are
{(−i )m+n+2s π Hm+s,n+s (iz, iz¯ ) : 0 ≤ m, n < N}.
k=1
r j + is j − rk − isk 2
1≤ j
2 2 e−(r j −x ) −(s j −y ) dr j ds j .
(3.42)
Hence the determinant formed by (−i )m+n (−1 )s Hm+s,n+s (iz, iz¯ ) : 0 ≤ m, n ≤ N is positive for N ≥ 0. Ismail [26] gave two extensions of the integral [45, Section 109, (4)],
Pn (x ) =
n!
2 √
π
∞
0
e−t t n Hn (xt )dt . 2
(3.43)
His two extensions of the above integral evaluation are in the following theorem. Theorem 3.9. With z = x + iy we have the integral representation
4
π
∞
0
∞
e−x
2
z (z¯ )m Hm,n (cz, cz¯ )dxdy
−y2 n
0 m n
= m!n!c c¯ 2 F1
−m, −n cc¯ − 1 , 1 cc¯
(3.44)
1
e−x
2
=
z¯ Hm2 ,n2 (z, z¯ )dxdy
−y2 m1 n1
z
R2
0
if m1 + m2 = n1 + n2 , or n2 > m1
m1 ! n1 !! ( m 1 − n 2 )!
if m1 + m2 = n1 + n2 .
(3.45)
Theorem 3.9 suggests considering the more general integrals
I ( m0 , m1 , . . . , mk ; n0 , n1 , . . . , nk ) 2 2 := π1 R2 e−x −y zm0 z¯n0 sj=1 Hm j ,n j (z, z¯ )dxdy,
z := x + iy.
Of course this integral does not have a closed form but has a combinatorial interpretation. Consider k + 1 two groups of disjoint sets S0 , S1 , . . . , Sk , T0 , T1 , . . . , Tk . The S sets are of a certain type, say type 1, and the T sets are of type 2. A perfect matching is a one to one mapping from ∪kj=0 S j onto ∪kj=0 T j . Further we allow the elements of S0 to be matched with elements of any T set, however elements of Sj , j = 0 are matched only with elements of Tr for r = j. Of course each element (vertex) is matched with a unique object. Theorem 3.10. The number of the above mentioned matchings is the integral in (3.46). 4. 2D − q-Hermite polynomials
where
ζ = ax + iby, a, b ∈ R, m = n + 2s, 2−2s π −m/2, (1 − m )/2 a2 − b2 C (m, s ) = . 2 F1 4 s+1 m! s!
N
|rk + isk |
(3.46)
see [25, (4.6.41)]. Another integral is
R2N
2s
j=1
π
Theorem 3.7. Let w = r + is with r, s ∈ R and z1 , z2 ∈ C, then we have the moment integral representation
1 π im+ n
N
and the integral evaluation
(3.36)
A two complex variable case is in [32].
e−z1 z2 Hm,n (z1 , z2 )=
×
Ismail and Zhang [33] introduced two q-analogues of the Ito polynomials. The first q-analogue is defined by
Hm,n (z1 , z2 |q ) :=
m ∧n
m k
k=0
It is clear that
q
n k
(−1 )k q(2 ) (q; q )k z1m−k z2n−k . k
(4.1)
q
Hm,n z2 , z1 q = Hn,m z1 , z2 q .
(4.2)
The second q-analogue is defined by the explicit representation
hm,n (z1 , z2 |q ) :=
m ∧n j=0
m j
q
n j
q(m− j )(n− j ) (−1 ) j (q; q ) j z1m− j z2n− j . q
(4.3)
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7
As a basic hypergeomoetric function (4.3) takes the form
Moreover they have the Rodrigues type formulas
hm,n (z1 , z2 |q )
Hm,n (z1 , z2 |q ) =
= (−1 )n qm−n+1 ; q z1m−n 1 φ1 n
q−n q
m−n+1
q; −qm+1 z1 z2 .
(1 − 1/q )m+n qmn m Dq−1 ,z2 Dnq−1 ,z1 ( (qz1 z2 ; q )∞ ), (qz1 z2 ; q )∞
(4.4)
Note the (m, n ) − (z1 , z2 ) symmetry
(4.19) n hm,n (z1 , z2 |q ) = (q − 1 )m+n (−z1 z2 ; q )∞ Dm q,z2 Dq,z1
hm,n (z1 , z2 |q ) = hn,m (z2 , z1 |q ),
(4.5)
and the fact
.
(4.20) Furthermore we also have the operational formula
hm,n (z1 , z2 |1/q ) = q−mn i−m−n Hm,n (iz1 , iz2 |q )
(4.6)
Hm,n (z1 , z2 |q ) = ((1 − q )2 Dq,z1 Dq,z2 ; q )∞ z1m z2n , qmn
Theorem 4.1. The polynomials {Hm, n (z1 , z2 |q)} and {hm, n (z1 , z2 |q)} have the generating functions
hm,n (z1 , z2 |q ) =
∞ ( uv; q )∞ um vn = Hm,n (z1 , z2 |q ) , (uz1 , vz2 ; q )∞ m,n=0 ( q; q )m ( q; q )n
and the lowering relations
1
(−z1 z2 ; q )∞
−q1/2 uz1 , −q1/2 vz2 ; q
(−uv; q )∞
∞
(4.7)
(
−q−1
(1 − q )
2D
)
q−1 ,z1 Dq−1 ,z2 ; q ∞
(4.21)
z1m z2n .
1 − qm Hm−1,n (z1 , z2 |q ), 1−q 1 − qn Dq,z2 Hm,n (z1 , z2 |q ) = Hm,n−1 (z1 , z2 |q ), 1−q Dq,z1 Hm,n (z1 , z2 |q ) =
∞ hm,n (z1 , z2 |q ) (m−n )2 /2 m n = q u v , (q; q )m (q; q )n m,n=0
(4.8) and satisfy the functional relations
Hm,n (qz1 , z2 |q ) = Hm,n (z1 , z2 |q ) − z1 (1 − qm )Hm−1,n (z1 , z2 |q ), (4.9)
(4.22)
and
qn−m+1 (1 − qm ) hm−1,n (z1 , z2 |q ), 1−q
Dq−1 ,z1 hm,n (z1 , z2 |q ) =
(4.23)
qm−n+1 (1 − qn ) Dq−1 ,z2 hm,n (z1 , z2 |q ) = hm,n−1 (z1 , z2 |q ). 1−q
We note that (4.11) and (4.12) imply the symmetry relation
Hm,n (z1 , qz2 |q ) = Hm,n (z1 , z2 |q ) − z2 (1 − qn )Hm,n−1 (z1 , z2 |q ), (4.10) Hm,n (qz1 , z2 |q )q
−m
Hm,n (qz1 , z2 |q )q−m = Hm,n (z1 , qz2 |q )q−n .
Theorem 4.2. The 2D − q-Hermite polynomials have the raising relations
Hm+1,n (z1 , z2 |q ) = qn
= Hm,n (z1 , z2 |q ) − q−1 (1 − qm )(1 − qn )Hm−1,n−1 (z1 , z2 |q ), (4.11)
Hm,n+1 (z1 , z2 |q ) = qm
Hm,n (z1 , qz2 |q )q−n (4.12)
hm,n z1 q−1 , z2 |q =hm,n (z1 , z2 |q ) + z1 (1−qm )q−m hm−1,n (z1 , z2 |q ), (4.13) hm,n z1 , z2 q−1 |q = hm,n (z1 , z2 |q ) + z2 (1 − qn )q−n hm,n−1 (z1 , z2 |q ), (4.14) q hm,n (z1 /q, z2 |q ) m
= hm,n (z1 , z2 |q ) + (1 − q )(1 − q )q m
n
1−m−n
1 − 1/q
(qz1 z2 ; q )∞ 1 − 1/q
Dq−1 ,z2 ( (qz1 z2 ; q )∞ Hm,n (z1 , z2 |q ) ),
(qz1 z2 ; q )∞
Dq−1 ,z1 ( (qz1 z2 ; q )∞ Hm,n (z1 , z2 |q ) ). (4.25)
= Hm,n (z1 , z2 |q ) − q−1 (1 − qm )(1 − qn )Hm−1,n−1 (z1 , z2 |q ).
(4.24)
hm−1,n−1 (z1 , z2 |q ), (4.15)
q hm,n (z1 , z2 /q|q )
and
(4.16) They also satisfy the three term recurrence relations
z1 Hm,n (z1 , z2 |q ) = qm (1 − qn )Hm,n−1 (z1 , z2 |q ) + Hm+1,n (z1 , z2 |q ), z2 Hm,n (z1 , z2 |q ) = qn (1 − qm )Hm−1,n (z1 , z2 |q ) + Hm,n+1 (z1 , z2 |q ). (4.17) and
qn z1 hm,n (z1 , z2 |q ) = hm+1,n (z1 , z2 |q ) + (1 − qn )hm,n−1 (z1 , z2 |q ), qm z2 hm,n (z1 , z2 |q ) = hm,n+1 (z1 , z2 |q ) + (1 − qm )hm−1,n (z1 , z2 |q ). (4.18)
hm,n (z1 , z2 |q ) hm+1,n (z1 , z2 |q ) = (q − 1 )(−z1 z2 ; q )∞ Dq,z2 , (−z1 z2 ; q )∞ (4.26) hm,n (z1 , z2 |q ) hm,n+1 (z1 , z2 |q ) = (q − 1 )(−z1 z2 ; q )∞ Dq,z1 . (−z1 z2 ; q )∞ Moreover the same polynomials have the multiplication formulas
Hm,n (az1 , bz2 |q ) =
m ∧n j=0
m j
q
n j
Hm− j,n− j (z1 , z2 |q ) (q, 1/ab; q ) j (q; q ) j , a j−m b j−n q
(4.27)
(q, ab; q ) j am− j bn− j hm− j,n− j (z1 , z2 ; q ).
(4.28)
hm,n (az1 , bz2 ; q )
n
= hm,n (z1 , z2 |q ) + (1 − qm )(1 − qn )q1−m−n hm−1,n−1 (z1 , z2 |q ),
=
m ∧n j=0
m j
q
n j
q
This theorem can be proved using the generating functions of Theorem 4.1. The polynomials Hm, n have the representation
Hm,n z1 , z2 q = (−1 )n
n (q; q )m q(2 ) m−n z1 pn z1 z2 , qm−n q , (q; q )m−n
(4.29)
where pn (x; qα |q) is the little q-Laguerre or Wall’s polynomials, [38], defined by
pn ( x; a|q ) =
2
φ1
q−n , 0 aq
q; qx .
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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Theorem 4.3. The polynomials orthogonality
Hm,n z, z q
satisfy the following
where ∞
dμ(z, z ) =
k=0
qk dθ δ (r − qk/2 ) , q ; q 2 π ( )k
dμ(z, z¯ ) ( u1 u2 v1 v2 ; q )∞ = . (u1 z, v1 z¯, v2 z, u2 z¯; q )∞ (q, u1 u2 , v1 v2 , u1 v1 , u2 v2 ; q )∞
The orthogonality relation of the h’s is given in the following theorem. Theorem 4.4. The polynomials {hm,n (z, z|q )} satisfy the orthogonality relation
hm,n (z, z|q )hs,t (z, z|q )
(−q1/2 u1 z, −q1/2 v1 z¯, −q1/2 v2 z, −q1/2 u2 z¯; q )∞ dxdy (−zz¯; q )∞ R2 (−u1 v1 , −u2 v2 , −u1 u2 , −v1 v2 ; q )∞ = π ln q−1 . ( u1 u2 v1 v2 ; q )∞
(4.33)
Theorem 4.5. The polynomials {Hm, n (z1 , z2 )} have the generating function
um (a/u; q )n vn (b/v; q )n Hm,n z1 , z2 q ( q; q )m ( q; q )n m,n=0 a/u, b/v (az1 .bz2 ; q )∞ = φ , q ; u v . (uz vz ; q ) 2 2
k
(4.36)
q
z1 , z2 q = z1m (1/z1 z2 ; q )m ,
(4.37)
(4.38)
The convergence in (4.36)–(4.38) is uniform on compact subsets C × C. The next two theorems give the Plancherel–Rotach asymptotics of the polynomials {Hm, n (z1 , z2 |q)} and {hm, n (z1 , z2 |q)}. Theorem 4.7. Let a, b, c, d ≥ 0, τ = τ (m, n; a, b, c, d ) = (a + c )m + (b + d )n and χ = χ (m, n; a, b, c, d ) = {(a + c )m + (b + d )n}, for 0 < τ (m, n; a, b, c, d) < m∧n, then
lim
m,n→∞
1 1 (q; q )∞ Hm,n z1 qam+bn− 4 , z2 qcm+dn− 4 |q (−z1 z2 )τ z1m z2n qam2 +(b+c )mn+dn2 −(m+n )/4−τ 2 /2−τ χ
= θ4 z 1 z 2 q χ ; q 1 / 2 , holds uniformly on compact subsets of the z1 and z2 planes, where ϑ4 is defined in (1.6).
az1 , bz2
hm,n w1 q−am−bn , w2 q−(1−a )m−(1−b)n |q (q; q )2∞ wm wn2 q(a−b) 1
mn−am2 +
(b−1 )
n2
1 w1 w2
.
Since the polynomial Hm,n (z, z¯ ) factors as a function of θ times a radial function it is clear that with z1 = z, z2 = z¯ the zeros of the polynomials investigated here as functions of z lie on circles. Now (4.38) implies the following asymptotic result. Theorem 4.9. Assume that the zeros of Hm,n (z, z¯|q ) lie on the circles with radii
r1 (H, m, n ) > r2 (H, m, n ) > · · · .
(4.40)
Then m,n→∞
(4.34)
= Aq
(4.39)
lim r j (H, m, n ) = q j/2 , j = 1, 2, . . . .
∞
lim z−n Hm,n n→∞ 2
m,n→∞
As it was the case with the Hm, n ’s the generating function (4.8) show that the orthogonality relation (4.32) is equivalent to the q-beta type integral
2
q(2 ) (−z1 z2 )−k = z2n (1/z1 z2 ; q )n ,
Similarly we establish the limiting relations
lim
where x = x + iy, x, y ∈ R, m, n, s, t ∈ N0 .
1
k=0
(4.32)
q
∞
n n k
Theorem 4.8. For a, b ∈ C and 0 < < 1 the following asymptotic result holds uniformly when w1 , w2 are in compact subsets of the complex plane
dxdy
(−zz; q )∞ π log q−1 (q; q )m (q; q )n = δm,s δn,t , (m−n )2 /2+(m+n )/2
R2
= z2n
m,n→∞
(4.31)
z1 , z2 q
It is clear that the orthogonality relation (4.30) and the generating function (4.7) imply the q-beta integral evaluation
C
lim z1−m z2−n Hm,n z1 , z2 q = (1/z1 z2 ; q )∞ .
and z = reiθ , r ∈ R+ , θ ∈ [0, 2π ], m, n, s, t ∈ N0 .
lim z−m Hm,n m→∞ 1
qmn (q; q )m (q; q )n Hm,n z, z q Hs,t z, z q dμ(z, z ) = δm,s δn,t , (q; q )∞ C (4.30)
[m5G;January 27, 2017;14:59]
(4.41)
Note that the limiting distribution of the radii of the circles of zeros coincides with the location of the masses of the orthogonality measure of the polynomials. Theorem 4.7 implies the following results about zeros of Hm, n .
As we saw in Sections 3 the Ito polynomials are orthogonal with respect to two different measures. The next theorem show that our Hm, n (z1 , z2 |q) share this property.
Theorem 4.10. The zeros of Hm,n z1 q(am+bn−1/4 ) , z2 q(cm+dn−1/4 ) |q
Theorem 4.6. We have the orthogonality relation
The function Aq (z) has infinity many zeros and they are all positive and simple. We denote them by
p s π j=0 k=0
×
0
(q, e2iθ , e−2iθ ; q )∞
0 < i1 ( q ) < i2 ( q ) < · · · < in ( q ) < · · · .
q−s q, q q1−p r 2
δs,p .
r1 (h, m, n ) > r2 (h, m, n ) > · · · . (4.35)
z, z q are straightforward.
(4.42)
Theorem 4.11. Assume that the zeros of hm,n (z, z¯|q ) lie on the circles with radii
H j,k (reiθ , re−iθ |q )Hs−k,p− j (reiθ , re−iθ |q ) dθ (q; q ) j (q; q )k (q; q )s−k (q; q ) p− j π r 2 p (1/r 2 ; q ) p = φ ( q; q ) p 1 1
asymptotically lie on the the curves z1 z2 qχ = q±(n+1/2 ) , n = 0, 1, 2, . . . . These curves become circles when z1 = z, z2 = z¯.
The large degree asymptotics of Hm,n Indeed (4.1), Tannery’s theorem, and the q binomial theorem show that
Then
lim q(m+n )/2 r j (h, m, n ) = 1/ i j (q ), j = 1, 2, . . . ,
m,n→∞
(4.43)
(4.44)
where {ij (q)} are the zeros of the Ramanujan function ordered as in (4.42).
Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001
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We now identify the q-Sturm–Liouville problems whose eigenfunctions are either {Hm, n (z1 , z2 |q)} or {hm, n (z1 , z2 |q)}. Using the first equations in (4.22) and (4.25) we find that Hm, n satisfy the eigenvalue equation
qz1 (1 − z1 z2 )Dq−1 ,z2 Dq,z1 f (z1 , z2 ) − Dq,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.45)
=
9
1 s 1 t ∞ hs,t (z1 , z2 |q ) q 2 emk q 2 e−mk (q; q )s (q; q )t
(4.55)
s,t=0
and when |z1 z3 | < 1, |z2 /z3 | < 1, we have
(z1 z2 ; q )∞
=
(−z1 z2 , z1 z3 , z2 /z3 ; q )∞
∞ hm,n (z1 , z2 |q ) (m+n )/2 (m−n )/2 q z3 . (q; q )m (q; q )n
m,n=0
(4.56)
with
λ = λm,n
1 − qm = −q1−n . ( 1 − q )2
(4.46)
5. The Hermite polynomials on Rn
Similarly we show that Hm,n solves the eigenvalue equation
We follow the notation in [12, Section 12.8]. First we define
qz2 (1 − z1 z2 )Dq−1 ,z2 Dq,z1 f (z1 , z2 ) − Dq,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.47)
φ (x ) = (Cx, x ),
with
λ = λm,n = −q1−m
1 − qn . ( 1 − q )2
(4.48)
Combine (4.23) and (4.26) to see that hm, n (z1 , z2 |q) satisfies the equation
(1 + z1 z2 )Dq,z2 Dq−1 ,z1 f (z1 , z2 ) −
z1 D −1 f (z1 , z2 )=λ f (z1 , z2 ), 1 − q q ,z1 (4.49)
where
λ = λm,n
1−q = qn−m+1 . ( 1 − q )2 m
(4.50)
z (1 + z1 z2 )Dq,z1 Dq−1 ,z2 f (z1 , z2 ) − 2 Dq−1 ,z1 f (z1 , z2 )=λ f (z1 , z2 ), 1−q (4.51) with
λ = λm,n =
1 − qn m−n+1 q . ( 1 − q )2
(4.52)
We note the relation m−n (m−n ) Ln n z1
hm,n (z1 , z2 |q ) = (−1 ) (q; q ) n
Rn
e−φ (x )/2+(a,x ) dx = κn eψ (a )/2 ,
j=1
m∈Nn0
n u j j=1
= e(Cx,u )−φ (u )/2 = eφ (x )/2−φ (x−u )/2 ,
Im,k := κn−1
δ (x − cqy ), y ∈ Z, c, α + 1 > 0,
m j!
= e(x,u )−ψ (u )/2 = eφ (x )/2−φ (x−C
n j=1
∂ m −φ ( x ) / 2 e , ∂x
m j and ∂∂ x := m
iθ
d σ re , re
−iθ
1 = d θ d μ ( r 2 ), 2
r ∈ R , θ ∈ [0, 2π ]
=
∞
1 s 1 t ∞ Hs,t (z1 , z2 |q ) (s−t )2 q 2 q 2 e2imk q 2 e−2imk , (q; q )s (q; q )t
Rn
−z1 z2 ; z1 e2mk , z2 e−2mk ; q
n j=1
m
∂ j . If m ∈ Z \ N , we set m 0 ∂x j j
n
m j !.
(5.6)
j=1
r Rn
Hmi (x )
i=1
s
Gki (x )e−φ (x )/2 dx,
i=1
(5.7)
{mi }ri=1
where ⊂ N0 and generating function
F (U, V ) := (4.54)
×
∞
{ki }si=1
{mi }ri=1 ⊂N0 {ki }si=1 ⊂N0
s,t=0
(z1 z2 ; q )∞
(5.4)
(5.5)
where r ∈
2
−z1 qe2imk , −z2 qe−2imk ; q
.
m ∈ Nn0 ,
Hm (x )Gk (x )e−φ (x )/2 dx = δm,k
I ({mi }ri=1 , {ki }si=1 ) := κn−1
Theorem 4.12. Assume that q = e−2k and |q| < 1, then we have
u )/2
More generally, Ismail and Simeonov [28] considered the integrals
+
is also an orthogonal measure for hm,n reiθ , re−iθ |q R+ , θ ∈ [0, 2π ].
−1
Hm (x ) = Gm (x ) = 0. The following biorthogonality relation holds [12, Section 12.8]:
etc. it is clear that our proof shows that
(5.3)
The second equality in (5.3) follows from the identity φ (x ) − 2(x, Cu ) + φ (u ) = φ (x − u ). The second equality in (5.4) is obtained by replacing u by C −1 u in (5.3). The generating function (5.3) leads to the Rodrigues formula [12, Section 12.8 (20)]
dμ(x ) = x−α wQL (x; α , c, λ )dx, x
m j! mj
Gm ( x )
where |m| =
(−x; q )∞
n u j
m∈Nn0
where Ln(α ) (z; q ) is a q-Laguerre polynomial, see (1.7). Since the moment problem associated with Ln(α ) (x; q ) is indeterminate [25, Section 21.8], they have infinitely many orthogonal measures. Let xα dμ(x) be such a measure, for example,
d μ (x ) =
(5.2)
mj
Hm (x )
(4.53)
α , λ, c > 0,
κn := (2π )n/2 −1/2
where at = (a1 , . . . , an ) ∈ Rn , dx = dx1 , . . . , dxn , and is the determinant of C, follows from diagonalizing φ and a change of variable on the normal distribution. Let m = (m1 , . . . , mn ) ∈ Nn0 . The polynomials {Hm (x)} and {Gm (x)} are defined through the generating functions [12, §12.8]
Hm (x ) = (−1 )|m| eφ (x )/2
(z1 z2 ; q ),
(5.1)
where C is a positive definite n × n matrix, xt = (x1 , . . . , xn ) ∈ Rn ,
and (x, y ) = nj=1 x j y j . The integral evaluation [12, §12.8]
Similarly we find that
ψ (x ) = (C −1 x, x ) = φ (C −1 x ),
s n i=1 j=1
i, j vm i, j
mi, j !
,
⊂ N0 . If F(U, V) is the exponential m
I ({
}
mi ri=1 ,
{ }
ki si=1
)
r n u i, j i, j i=1 j=1
mi, j ! (5.8)
where U = [ui, j ] is a r × n matrix and V = [vi, j ] is an s × n matrix.
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A formula of Feldheim [13] is
i−|m| Hm (ix ) = κn−1
n
Rn
(Cy )
mj j
e−φ (x−y )/2 dy,
m ∈ Nn0 .
j=1
(5.9) Burchnal [7] proved that
d −
dx
+ 2x
m
f (x ) =
m
m Hm−k (x ) f (k ) (x ). k
(−1 )k
k=0
(5.10)
Using a technique similar to Louck’s proof of the Kibble–Slepian formula, [41], Ismail and SImeonov also proved the following generating function. Theorem 5.2. Let S and C be positive definite n × n matrices such that I + 12 C 1/2 SC 1/2 is invertible, and let {Hm (x)} be the Hermite polynomials associated with C. Then
−1/2
(s /2 )m j,l j,l −tr(M )
× exp (1/4 ) C SC =
2
1/2
M∈Mn (N0 ) M =Mt
I +C
1/2
SC
1/2
/2
−1
x, x
Hm (x ),
m j,l !
1≤ j≤l≤n
C
1/2
∞
mj
m j!
[ψ (a )/2]m/2 Hm m!
Gm1 ,m2 ,··· ,mn (x1 , . . . , xn )
( a, x ) . [2ψ (a )]1/2
(5.16)
φm (r; α )φn (r; α )rα dμ(r, β ) = ζn (α )δm,n , α ≥ 0.
0
n
det I + C 1/2 SC 1/2 /2
(5.15)
We introduced the model of this section in [31]. Let {φ n (r; α )} be a system of orthogonal polynomials satisfying the orthogonality relation
−
1
(Ca, x ) , [2φ (a )]1/2
6. Complex 2D-systems
Theorem 5.1.
m j ∂ + (Cx ) j f (x ) ∂xj j=1 m1 mn n mj ∂k = ··· (−1 )|k| Hm−k (x ) f ( x ). ∂x kj j=1 k =0 k =0
m a j
m1 +m2 +···+mn =m j=1
=
Ismail and Simeonov [28] extended proved the following multivariate version of Burchnal’s formula. n
[φ (a )/2]m/2 Hm m!
=
It is assumed the measure μ does not depend on α but may depend on another parameter β . Let
φn (r; α ) =
n
c j (n, α )r n− j ,
c j (n, α ) ∈ R,
(6.2)
j=0
and define bivariate polynomials fm,n by
fm,n (z1 , z2 ; β ) =
z1m−n φn (z1 z2 ; m − n + β ), m ≥ n, fn,m (z2 , z1 ; β ), m < n.
∂ f (z, z¯; β ) = (m − n ) fm,n (z, z¯; β ), m ≥ n. ∂θ m,n
Therefore the functional relations
(5.11)
(δz1 − δz2 ) fm,n (z1 , z2 ; β ) = (m − n ) fm,n (z1 , z2 ; β ),
m ≥ n, (6.4)
1 − qm−n fm,n (z1 , z2 ; β ), δq,z1 − qm−n δq,z2 fm,n (z1 , z2 ; β ) = 1−q
m ≥ n,
(6.5)
Moreover Ismail and Simeonov [28] proved that Proposition 5.3. Let S = {S j }nj=1 be a multiset of n sets, where each set Sj contains m j = |S j | distinct objects, j = 1, . . . , n. Let M(m ) be the set of all inhomogeneous perfect matchings of this multiset. Then
Hm (x ) =
(−cedge )
μ∈M(m ) edge∈E (μ )
fix∈Fix(μ )
(Cx )i(fix) ,
(5.12)
where E(μ) denotes the set of all edges and Fix(μ) denotes the set of all fixed points (unmatched elements) in a matching μ ∈ M(m ), cedge = c j1 , j2 , if edge ∈ E(μ) connects elements from sets S j1 and S j2 , and i(fix ) = j if fix ∈ Sj .
With the notation m = nj=1 m j , the polynomials {Hm (x) and {Gm (x) have the Rodrigues type representations
Hm (x ) = (−1 )m exp(φ (x/2 ))
∂ exp(−φ (x/2 ), n 1 ∂ xm , . . . , ∂ xm n 1 m
(5.13) Gm (x ) = (−1 )m exp(ψ (x/2 ))
∂
∂m 1 xm ,..., 1
n ∂ xm n
exp(−ψ (x/2 ). (5.14)
One curious property of the Hermite polynomials is the following
mj
m a j
m1 +m2 +···+mn =m j=1
m j!
Hm1 ,m2 ,...,mn (x1 , . . . , xn )
(6.3)
From (6.3) it is clear that
−i
where Mn (N0 ) is the set of all n × n matrices with entries from
N0 , M = [m j,l ]nj,l=1 , tr(M ) = nj=1 m j, j is the trace of M, and m =
(m1 , . . . , mn ) with m j = m j, j + nl=1 m j,l , j = 1, . . . , n.
(6.1)
hold. Theorem 6.1 [31]. The polynomials {fm, n (z1 , z2 ; β )} satisfy the orthogonality relation
dθ d μ (r 2 ; β ) 2π = ζm∧n (|m − n| + β )δm,s δn,t ,
R2
fm,n (z, z¯; β ) fs,t (z, z¯; β )
(6.6)
for all nonnegative integers m, n, s, t, where dμ(r; β ) = r β dμ(r ). The proof follows from (6.1). Unlike general 2D orthogonal polynomials [11] the fm, n ’s satisfy three term recurrence relations. Indeed we have
z2 fm+1,n (z1 , z2 ; β ) =
c0 (n, m − n + 1 + β ) fm+1,n+1 (z1 , z2 ; β ) c0 ( n + 1 , m − n + β )
c0 (n, m − n + 1 + β )cn+1 (n + 1, m − n + β ) fm,n (z1 , z2 ; β ), c0 (n + 1, m − n + β )cn (n, m − n + β ) m ≥ n, (6.7) −
and
z1 fm,n (z1 , z2 ; β ) −
c0 (n, m − n + β ) fm+1,n (z1 , z2 ; β ) c0 (n, m + 1 − n + β )
= um,n fm,n−1 (z1 , z2 ; β ),
(6.8)
with
um,n =
c0 (n, α + 1 )c1 (n, α ) − c0 (n, α )c1 (n, 1 + α ) c0 (n − 1, 1 + α )c0 (n, 1 + α )
(6.9)
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and α = m − n + β . Assume that the three term recurrence relation satisfied by the φ n ’s is
r φn (r ; α )=an (α )φn+1 (r; α ) + cn (α )φn (r; α ) + bn (α )φn−1 (r; α ) (6.10) By comparing like powers of r we find that
c0 (n, α ) an ( α ) = , c0 ( n + 1 , α )
c1 ( n + 1 , α ) c1 (n, α ) cn ( α ) = − , c0 (n, α ) c0 ( n + 1 , α ) (6.11)
11
where
δq,r f = rDq,r f.
(6.21)
Then fm, n (z1 , z2 ), for m ≥ n and α = m − n + β , satisfies the q-partial differential equations 2 An (z1 z2 , α )δq,z f + Bn (z1 z2 , α )δq,z2 f + Cn (z1 z2 , α ) f = 0, 2
(6.22)
2 An (z1 z2 , α )δq,z f + {qα Bn (z1 z2 , α ) − 2[α ]q An (z1 z2 , α )}δq,z1 f 1
+ [α ]2q An (z1 z2 , α ) − [α ]q qα Bn (z1 z2 , α ) + q2α Cn (z1 z2 , α ) f = 0. bn ( α ) =
c0 (n, α )c2 (n, α ) − c1 (n, α )2 c0 (n − 1, α )c0 (n, α ) c0 (n, α )c2 (n + 1, α ) − c1 (n, α )c1 (n + 1, α ) − c0 ( n − 1 , α )c0 ( n + 1 , α )
(6.12)
Theorem 6.5. Assume that
wα +β ( x ) = xα wβ ( x )
The following recurrence relation follows from (6.2) and (6.3).
(6.13)
We note that the definition of the polynomials fm, n indicates that for m ≥ n it may have a trivial zero at z1 = 0 but will vanish on the curves z1 z2 = r where r is a zero of φn (r; m − n + β ). The zeros of φn (r; m − n + β ) are positive, real and simple when m ≥ n. The relationships (3.9), (4.29), and (4.53) show that the polynomials Hm, n and their q-analogues are examples of fm, n polynomials. Remark 6.2. In view of the defining Eq. (6.3) the question of finding the large m, n asymptotics of the two variate polynomials fm, n is equivalent to finding the large m, n behavior of φn (r; m − n + β ). It will be of interest to carry out this program at least for some special systems, including the Freud type polynomials orthogonal with respect to xα exp(−p(x )), where p is a polynomial with positive leading term. Theorem 6.3. Let φ n (x; α ), an (α ), bn (α ) as in (6.11) and (6.12), for m ≥ n − 1 we have
z1 fm,n (z1 , z2 ) − vm,n fm+1,n (z1 , z2 ) = um,n fm,n−1 (z1 , z2 ),
(6.14)
(6.24)
and
φn ( x ; α ) =
(z1 z2 − cn (m − n + β )) fm,n (z1 , z2 ; β ) = an (m − n + β ) fm+1,n+1 (z1 , z2 ; β ) + bn (m − n + β ) fm−1,n−1 (z1 , z2 ; β ).
1 ∂ n (wα (x )xn ), wα ( x ) x
vm,n = an (m − n ).
(6.15)
We next consider differential or q-difference equations satisfied by fm, n (z1 , z2 ). Theorem 6.4. Assume that φ n (r; α ) satisfies the second order differential equation
An (r, α )δr2 f + Bn (r, α )δr f + Cn (r, α ) f = 0,
(6.16)
fm,n (z1 , z2 ; β ) =
z m−n ∂ n w (z1 z2 )zm β z2 2 1 , z2 wβ ( z1 z2 ) ∂z2
fm+1,n+1 (z1 , z2 ; β ) =
m+1−n z2 z1
(6.26)
wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
z2 m−n z1
wβ ( z1 z2 )
,
(6.27)
∂zn1 wβ (z1 z2 )z1m fm,n (z1 , z2 ; β ) = , wβ ( z1 z2 ) ∂z1 wβ (z1 z2 ) fm+1,n (z1 , z2 ; β ) fm+1,n+1 (z1 , z2 ; β ) = wβ ( z1 z2 ) and
∂z2
m+1−n z 2
z1
(6.28)
(6.29)
wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
(6.30)
We next state a q-analogue of the previous result. Theorem 6.6. Assume that
wα +β ( x ) = wα +β ( x|q ) = xα wβ ( x|q )
(6.31)
and
φn (x; α ) = φn (x; α|q ) =
1 Dn (wα (x|q )xn ), wα (x ) q,x
n = 0, 1, . . . . (6.32)
∂ δr f = r f. ∂r
(6.17)
Then, fm, n (z1 , z2 ; β ), for α = m − n + β and m ≥ n, satisfies the second order partial differential equations
An (z1 z2 , α )δz22 f + Bn (z1 z2 , α )δz2 f + Cn (z1 z2 , α ) f = 0, An (z1 z2 , α )δz21 f + {Bn (z1 z2 , α ) − 2α An (z1 z2 , α )}δz1 f +
(6.25)
Then for m ≥ n we have
where
n = 0, 1, . . . .
= ∂z1 wβ (z1 z2 ) fm+1,n (z1 , z2 ; β )
where
um,n = bn (m − n ),
(6.23)
α 2 An (z1 z2 , α ) − α Bn (z1 z2 , α ) + Cn (z1 z2 , α ) f = 0.
fm,n (z1 , z2 ; β|q ) =
z1 m−n z2
n ∂q,z wβ (z1 z2 |q )z2m 2 , wβ ( z1 z2 |q )
(6.33)
(6.18) fm+1,n+1 (z1 , z2 ; β|q ) (6.19)
Similarly, assume that φ n (r; α ) satisfies the second order q-difference equation 2 An (r, α )δq,r f + Bn (r, α )δq,r f + Cn (r, α ) f = 0,
Let fm,n (z1 , z2 ; β|q ) = fm,n (z1 , z2 ; β ), then for m ≥ n we have
(6.20)
=
∂q,z2
m+1−n z2 z1
wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q )
z2 m−n z1
wβ ( z1 z2 |q )
n ∂q,z wβ (z1 z2 |q )z1m 1 fm,n (z1 , z2 ; β|q ) = , wβ ( z1 z2 |q )
,
(6.34)
(6.35)
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fm+1,n+1 (z1 , z2 ; β|q ) =
∂q,z1 wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q ) wβ ( z1 z2 |q )
and
and
∂q,z2
m+1−n z 2
z1
z m−n 2
=
z1
is the little q-Jacobi (α ) Rl,m z, z∗; q2 satisfy α
(6.37)
∂q,z1 wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q ) .
1 2π
k=0
n k!(−1 )k (ν )m+n−k z1m−k z2n−k , k
ν > −1. (7.38)
m,n
( z1 , z2 ) = ( ν )
m n m+n z1 z2 2 F1
−m, −n 1 −m − n − ν + 1 z1 z2
(7.39)
which a constant multiple of the disk polynomials of Section 2.3 in [11]. They have the generating function ∞
um vn = (1 − uz1 − vz2 + uv )−ν , m,n (z1 , z2 ) ( m! n!
Cν
m,n=0
(7.40)
whose proof is an exercise in the application of the binomial theν (z , z ) orem. Next we solve the connection relation between Cm,n 1 2 and Hm, n (z1 , z2 ). We claim that
Cν
m,n
m ∧n
( z1 , z2 ) =
( ν )m+n m! n! Hm−p,n−p (z1 z2 ) 1 p! (m − p)! (n − p)!
p=o
× F1 (−p; −ν − m − n + 1; −1 ) t ν −1
(ν )
0
=
∞
e
(ν )
0 ∞
=
t ν −1
r,s=0
r,s=0
j,k=0
Equating coefficients of um vn we see that m = r + j + k, n = s + j + k. Let p = j + k. Now (7.41) follows after some manipulations. It is clear from the generating function (7.40) that the disk polynomials have the convolution property ν +μ
Cm,n (z1 , z2 ) =
m n
μ
ν C j,k (z1 , z2 )Cm − j,n−k (z1 , z2 )
(7.42)
j=0 k=0
Floris [16], see also [15] and [17], introduced the following qanalogue of the disk polynomials. For α , β > −1 and l, m ∈ Z+ , the (α ) Floris q-disk polynomials Rl,m z, z∗; q2 are defined by (α )
Rl,m z, z∗; q
2
=
α ,l−m )
zl−m Pm(
(α ,m−l )
Pm
1 − zz∗ : q2
∗ m−l
1 − zz : q (z ) ∗
2
l≥m l ≤ m,
where ∗
0
α
∗
α
( ) iθ Rl,m e z, e−iθ z∗; q2 Rl( ,m) eiθ z, e−iθ z∗; q2 dθ
1 − q2
q2 ; q2
1 − q2(α +l+m+1 )
l
q2 ; q2
m
q2m(α +1 ) δl l δmm
q2(α +1 ) ; q2
l
q2(α +1 ) ; q2
, m
(k ) n q 2 (q; q )k (bq; q )m+n−k , (7.43) k (−1 )k z1k−m z2k−n q q k=0
∞ m pm,n (z1 , z2 ; b|q ) = k
it is clear that
pm,n (z2 , z1 ; b|q ) = pn,m (z1 , z2 ; b|q )
(7.44)
then for m ≥ n we have
pm,n (z1 , z2 ; b|q ) = (−1 )n q(2 ) (bq; q )m qm−n+1 ; q z1m−n n n
× pn z1 z2 ; qm−n , b|q ,
(7.45)
where pn (x; a, b|q) is the little Jacobi polynomials. Theorem 7.1. For 0 < q < 1 and b < q−1 , the polynomials { pm,n (z, z; b|q )} satisfy the following orthogonality
pm,n (z, z; b|q ) ps,t (z, z; b|q )dμ(z, z )
(bq; q )∞ qmn (q, bq; q )m (q, bq; q )n δm,s δn,t , 1 − bqm+n+1 (q; q )∞
where
(7.46)
(bq; q ) qk dθ k δ r − qk/2 , 2π (q; q )k ∞
(7.47)
z = reiθ , r ∈ R+ , θ ∈ [0, 2π ] and m, n, s, t ∈ N0 .
ur vs r+s Hr,s (z1 z2 ) t dt r! s!
∞ ur vs (−1 ) j (uv ) j+k (ν + r + s + j + 2k ) . r! s! j! k! (ν )
Hr,s (z1 z2 )
polynomials
k=0
∞
−t +t (t −1 )uv
0
2π
dμ(z, z ) =
e−t +t uz1 +t vz2 −t uvdt
q-disk
for α We now introduce our q-analogue of these polynomials. For 0 < q < 1 and b < q−1 , let us define
= (7.41)
The
> −1, l , l , m, m ∈ Z+ .
C
Write the right-hand side of (7.40) as ∞
1
They are also known as the disk polynomials or the Zernike polynomials, [11]. It is clear that
Cν
=
m k
α
( ) = Rm,l z, z∗; q2
× (1 − zz∗ )α dq2 (1 − zz∗ )
The 2D-ultraspherical polynomials are ν (z , z ) = Cm,n 1 2
∗
polynomials.
and the orthogonality relation
7. 2D q-ultraspherical polynomials
m ∧n
( ) Rl,m z, z∗; q2
wβ (z1 z2 |q ) fm+1,n (z1 , z2 ; β|q )
( x : q ) = pm x; qα , qβ ; q
α ,β )
Pm( (6.36)
[m5G;January 27, 2017;14:59]
Theorem 7.2. The polynomials {pm, n (z1 , z2 ; b|q)} have the generating function ∞ pm,n (z1 , z2 ; b|q ) m n u v (q; q )m (q; q )n
m,n=0
uz1 , vz2 (bq, uv; q )∞ = 2 φ1 uz , z v ; q uv ( 1 2 )∞
q; bq .
(7.48)
For bq, cq < 1 and b = 0, the connection relation between the q − 2D ultraspherical polynomials is given by
pm,n (z1 , z2 ; b|q ) = (bq; q )∞
∞ j=0
c
;q b
j
( q; q ) j
bq
m+n 2 +1
j
pm,n z1 q 2 , z2 q 2 ; c q j
j
(cq; q )∞
.
(7.49) The connection relation between the q − 2D ultraspherical and q − 2D Hermite is given by
∞ (m+n )/2+1 pm,n (z1 , z2 ; b|q ) bq = (bq; q )∞ ( q; q ) j
j
Hm,n z1 q j/2 , z2 q j/2 |q .
j=0
2
∗
2
z z = q zz + 1 − q ,
(7.50)
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Moreover we have the inverse relation
Hm,n (z1 , z2 |q ) =
∞ −bq(m+n )/2+1 k=0
k
(bq; q )∞ (q; q )k
q(2 ) pm,n z1 qk/2 , z2 qk/2 ; b|q . k
(7.51)
2
φ1 =
uz1 , vz2 uv
1 − qm−n pm+1,n+1 (z1 , z2 ; b|q )
= z1 1 − bqm+1 − z2 q
m−n
− z1 1 − bq
∞ (uz1 , z2 v; q )∞ pm,n (z1 , z2 ; b|q ) m n u v . (bq, uv; q )∞ m,n=0 (q; q )m (q; q )n
(7.52)
1 − qm+1 pm,n+1 (z1 , z2 ; b|q )
n+1
1−q
n+1
(7.64)
pm+1,n (z1 , z2 ; b|q ),
pm+1,n+1 (z1 q, z2 ; b|q ) − pm+1,n+1 (z1 , z2 q; b|q ) = z2 1 − bqm+2
q; bq
1 − bq
Let us rewrite (7.48) in the form
13
m+1
1 − qn+1 pm+1,n (z1 q, z2 ; b|q )
1−q
m+1
(7.65)
pm,n+1 (z1 , z2 q; b|q ),
z2 (1 − qn ) pm,n−1 (z1 q, z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 q; b|q ) = z2 (1 − qn ) pm,n−1 (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ), (7.66)
Theorem 7.3. The polynomials {pm, n (z1 , z2 ; b|q)} satisfy the following properties,
Dq,z1 pm,n (z1 , z2 ; b|q ) =
(1 − bq ) 1−q
(1 − qm ) pm−1,n (z1 , z2 ; bq|q ),
qz1 z2 ( pm,n (z1 q, z2 ; b|q ) − pm,n (z1 , z2 q; b|q )) + z1 z2 (1 − qm )(1 − qn )( pm−1,n−1 (z1 q, z2 ; b|q ) − pm−1,n−1 (z1 , z2 q; b|q ))
(7.53) Dq,z2 pm,n (z1 , z2 ; b|q ) =
(1 − bq ) 1−q
(1 − q ) pm,n−1 (z1 , z2 ; bq|q ), n
(7.54)
(qz1 z2 ; q )∞ Dq−1 ,z1 pm,n (z1 , z2 ; b|q ) (bqz1 z2 ; q )∞ (qz1 z2 ; q )∞ pm,n+1 z1 , z2 ; bq−1 |q = , qm−1 (q − 1 )(bz1 z2 ; q )∞ (qz1 z2 ; q )∞ Dq−1 ,z2 pm,n (z1 , z2 ; b|q ) (bqz1 z2 ; q )∞ (qz1 z2 ; q )∞ pm+1,n z1 , z2 ; bq−1 |q = , qn−1 (q − 1 )(bz1 z2 ; q )∞
(7.56)
(7.57) pm,n (z1 q, z2 ; b|q ) − bq2m−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 q; b|q ) = pm,n (z1 , z2 ; b|q ) − qm−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ), (7.58) pm,n (z1 , z2 q; b|q ) − bq
(1 − q )(1 − q ) pm−1,n−1 (z1 q, z2 ; b|q ) n
= qn pm,n (z1 , z2 ; b|q ) − qn−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ), (7.59) pm,n (z1 , z2 q; b|q ) − bqm+n (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 q; b|q ) = qn pm,n (z1 , z2 ; b|q ) − qn−1 (1 − qm )(1 − qn ) pm−1,n−1 (z1 , z2 ; b|q ). (7.60) pm,n (z1 q, z2 ; b|q ) − bqn+1 z1 (1 − qm ) pm−1,n (z1 q, z2 ; b|q ) = pm,n (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ), pm,n (z1 q, z2 ; b|q ) − bqm z1 (1 − qm ) pm−1,n (z1 , z2 q; b|q ) = pm,n (z1 , z2 ; b|q ) − z1 (1 − qm ) pm−1,n (z1 , z2 ; b|q ),
+ z1 (1 − qm )( pm−1,n (z1 q, z2 ; b|q ) − pm−1,n (z1 q, z2 q; b|q )) − z2 (1 − qn )( pm,n−1 (z1 , z2 q; b|q ) − pm,n−1 (z1 q, z2 q; b|q )). The inversion transformation of quanta q → q−1 in (4.6) relates the properties of one family of polynomials for q > 1 to the properties of another family of polynomials with 0 < q < 1. The polynomials pm, n (z1 , z2 ; b|q) are essentially invariant under the quanta inversion transformation,
pm,n z1 , z2 ; b|q−1
= pm,n (z1 , z2 ; b|q )qm − qm−1 (1−qm )(1−qn ) pm−1,n−1 (z1 , z2 ; b|q ),
m
− qz12 z2 (1 − qm )( pm−1,n (z1 , z2 q; b|q ) − bpm−1,n (z1 q, z2 q; b|q ))
(7.67) (7.55)
pm,n (z1 q, z2 ; b|q ) − bqn+m−1 (1−qm )(1−qn ) pm−1,n−1 (z1 q, z2 ; b|q )
2n−1
= qz1 z22 (1 − qn )( pm,n−1 (z1 q, z2 ; b|q ) − bpm,n−1 (z1 q, z2 q; b|q ))
m+n
= (−b)
∞ m k k=0
q
n k
(−b)−k (q; q )k
m+n+1 q ( 2 )
m+n −
= (−b) ×
q b
;q
∞
k+1 2
m+n−k
;q
m+n−k
) zm−k zn−k 1 2 q k k n q (2 ) − ( q; q )
m+n−k+1 2
k=0
b
q
×qk(k−m )+k(k−n )−( )−( )−( k 2
q
m k
q
k
k
b
q
z1m−k z2n−k
m+n q ( 2 ) pm,n 1 α 1 −α × (b/q ) z1 , (b/q ) z2 ; |q . b
= (−1 )m+n (b/q )(
1−α )m+α n −
Therefore, we have established the symmetry
pm,n z1 , z2 ; b|q−1
=
bq−1
(1−α )m+αn m+n
(−1 )
q(
m+n 2
)
α
1 −α
pm,n (b/q ) z1 , (b/q )
z2 ; 1/b|q ,
(7.68)
(7.61)
for α ∈ C. We now come the asymptotics of pm,n (z1 , z2 ; b|q).
(7.62)
Theorem 7.4. Let z1 , z2 ∈ C, bq < 1 and z1 z2 = 0, then we have
lim
m,n→∞
1 + bqm+n z1 pm,n (z1 , z2 ; b|q ) = z1 pm−1,n (z1 , z2 ; b|q ) − qm−1 (1 − qn )(1 − bqn ) pm−1,n−1
pm,n (z1 , z2 ; b|q ) = (bq; q )∞ z1m z2n
1 z1 z2
;q
∞
,
(7.69)
uniformly on compact subsets of the z1 and z2 planes.
× (z1 , z2 ; b|q ), (7.63)
The theorem follows from the definition (7.43) and Tannery’s theorem.
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(β )
8. The polynomials {Zm,n (z1 , z2 )}
(β ) ) (nδz1 − mδz2 )Zm,n (z1 , z2 ) = (m − n )(m + β )Zm(β−1 ( z , z ), ,n−1 1 2
Motivated by the class of general 1D systems in Section 2 we (β ) define polynomials {Zm,n (z1 , z2 )} by
n 1 n ( β + 1 )m (β ) Zm,n (z1 , z2 ) = (−1 )n−k z1m−k z2n−k , n! k (β + 1 )m−k
(8.1)
k=0
for m ≥ n. When m < n the polynomials are defined by (β )
(β )
Zm,n (z1 , z2 ) = Zn,m (z2 , z1 )
Here β > −1. These polynomials arise through the choice φn (x; α ) = ( α +β ) Ln (x ), where Ln(a) (x ) is a Laguerre polynomial, [25,50], ) n n Ln(a ) (x ) = (a+1 k=0 n!
(−x )k k (a+1 )k
)n n = (a+1 k=0 n!
n
(−x )n−k . k (a+1 )n−k
Zm,n (z1 , z2 ) =
(β +m−n ) z1m−n Ln
( z1 z2 )
z1 Zm,n
(8.3)
(β )
(z1 , z2 ) = Zm+1,n (z1 , z2 ).
(α + β + n + 1 )
(8.6)
(β )
Zm,n (z, z¯ ) Zs,t (z, z¯ ) (zz¯ )β e−zz¯ rdrdθ
(β )
(8.8)
(β +1 )
) (z1 , z2 ) = −Zm(β+1 (z , z ) ,n 1 2
(8.20)
1 (β ) (β ) ∂z wβ (z1 z2 )Zm,n (z1 , z2 ) = (n + 1 )Zm,n (z , z ) +1 1 2 wβ ( z1 z2 ) 1
δz j wβ (z1 z2 ) = (β − z1 z2 )wβ (z1 z2 ),
j = 1, 2
(8.22)
and (8.8) which imply the relationships
(β ) δz2 wβ (z1 z2 )Zm,n ( z1 , z2 )
(β )
(β )
= β Zm,n (z1 , z2 ) − z2 Zm+1,n (z1 , z2 ), (8.23)
wβ ( z1 z2 ) (β )
z2 Zm+1,n (z1 , z2 ) = −(n + 1 )Zm+1,n+1 (z1 , z2 )
(β )
(8.24)
(β )
(β )
+ (β + m + 1 )Zm,n (z1 , z2 ),
(8.9)
and
Theorem 8.3. The polynomials {Zm,n (z1 , z2 )} satisfy the second order partial differential equation
∂∂ z1 ∂∂ z2 f +
(β ) ( z1 , z2 ) (β + m + n + 1 − z1 z2 )Zm,n (β )
(β ) δz1 wβ (z1 z2 )Zm,n ( z1 , z2 ) = (β + m − n )Zm,n (z1 , z2 ) − z2 Zm+1,n (z1 , z2 ).
(β )
(β )
= (n + 1 )Zm+1,n+1 (z1 , z2 ) + (m + β )Zm−1,n−1 (z1 , z2 )
(8.10)
respectively, where m ≥ n. We now discuss differential recurrence relations. (β )
Theorem 8.2. For m ≥ n, the polynomials {Zm,n (z1 , z2 )} satisfy the differential recurrence relations (β ) (β ) δz2 Zm,n (z1 , z2 ) = −z2 Zm,n ( z , z ), −1 1 2
(8.11)
(β ) (β ) ) δz2 Zm,n (z1 , z2 ) = nZm,n (z1 , z2 ) − (β + m )Zm(β−1 ( z , z ), ,n−1 1 2
(8.12)
(β ) (β ) (β ) δz1 Zm,n (z1 , z2 ) = (m − n )Zm,n (z1 , z2 ) − z2 Zm,n ( z , z ), −1 1 2
(8.13)
(β ) (β ) ) δz1 Zm,n (z1 , z2 ) = mZm,n (z1 , z2 ) − (m + β )Zm(β−1 (z , z ), (8.14) ,n−1 1 2 (β )
(β )
and
z1 Zm,n (z1 , z2 ) = Zm+1,n (z1 , z2 ) − Zm,n−1 (z1 , z2 ), (β )
(8.7)
In view of (8.6) the recurrence relations (6.8), (6.7) and (6.13) become (β )
(8.19)
It is clear that (8.18) implies the differentiation formulas
wβ ( z1 z2 )
(β + m ∨ n + 1 ) δm,s δn,t . ( m ∧ n )!
(β )
(8.18)
for m ≥ n. It also clear that
Theorem 8.1. For m, n, s, t = 0, 1, . . . and β > −1, we have the orthogonality relation
=π
(8.21)
, n! n− j (−1 ) (α + β + 1 )n c j (n, α ) = . ( n − j )! j ! ( α + β + 1 ) n − j
(β )
(8.17)
2
1 −β z1 z2 n m+β −z1 z2 z e ∂z1 z 1 e n! 1 (−1 )m = (z1 z2 )−β ez1 z2 ∂zn1 (z1 z2 )β ∂zm2 e−z1 z2 . n!
(β )
Zm,n (z1 , z2 ) =
and
(8.5)
Therefore in the notation of Section 2, we have
R2
β +m n z
(8.4)
(0 ) Hm,n (z1 , z2 ) = (−1 )n (m ∧ n )!Zm,n ( z1 , z2 ).
−β
z1 exp(−∂z1 ∂z2 )z1
ez1 z2 ∂z2 e−z1 z2 Zm,n (z1 , z2 ) = −z1 Zm,n
It is clear that when β = 0 we see that
ζn (α ) =
n!
wβ (z1 z2 ) = (z1 z2 )β exp(−z1 z2 ).
and (β +1 )
(−1 )n
(β )
Zm,n (z1 , z2 ) =
We shall use the notation
Indeed, for m ≥ n (β )
where δz1 = z1 ∂z1 , δz2 = z2 ∂z2 . Moreover they have the operational representation
and the Rodrigues type representation
(8.2)
n
(8.16)
(β )
(δz1 − δz2 )Zm,n (z1 , z2 ) = (m − n )Zm,n (z1 , z2 ),
(8.15)
β − z1 z2 z1
∂∂ z2 f = −n f,
(8.25)
for all m ≥ n. It is clear that the differential property (8.25) can be written in the form
∂ ∂ (β ) (β ) w (z z ) Z (z , z ) = −nZm,n (z1 , z2 ). ∂ z1 β 1 2 ∂ z2 m,n 1 2
(8.26)
Note that (8.25) and (8.15) indicate that the polynomials (β ) {Zm,n (z1 , z2 )} are simultaneous eigenfunctions of the operators on their respective left-hand sides. The differential operators β −z1 z2
∂∂ z1 ∂∂ z2 +
z1
∂∂ z2 and z1 ∂∂ z1 − z2 ∂∂ z2 indeed commute as
can be directly verified. Let w(x, y ) = (zz¯ )β e−zz¯ and define the inner product
< f, g >=
R2
f (x, y )g(x, y )w(x, y )dxdy.
(8.27)
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( β ,γ )
9. The polynomials {Mn
Theorem 8.4. We have the adjoint relations
(∂z )∗ = −∂z¯ −
β
+ z,
z¯
(∂z¯ )∗ = −∂z −
β z
+ z¯.
(8.28)
The proof is straightforward calculus exercise. This allows us to write (8.25) or (8.26) in the selfadjoint form
( (∂z¯ )∗ ∂z¯ ) f = n f.
15
( z1 , z2 )}
For α > 0, β , γ > −1 Ismail and Zeng [29] introduced the poly( α + γ ,β ) nomials φn (r; α ) = Pn (1 − 2r ), that is
φn (r; α ) = (α + γ + 1 )n
(8.29)
n (α + β + γ + n + 1 )n−k (−r )n−k . k!(n − k )!(α + γ + 1 )n−k k=0
(9.1)
Moreover we also have the following result. Theorem 8.5. The operator A := (∂z¯ )∗ ∂z¯ is positive in the sense that < Af, f > ≥0 with = if and only if ∂z¯ f = 0, that is f depends only on z.
They satisfy the following orthogonality relation
1 0
φm (r; α )φn (r; α )uα+γ (1 − u )β du = ζn (α + γ , β ) δm,n , (9.2)
where
This indicates that
1 ∂ (β ) (β ) w(x, y )Zm,n (z, z¯ ) = (n + 1 )Zm,n (z, z¯ ). w(x, y ) ∂ z
(8.30)
ζn (α + γ , β ) =
(α + γ + n + 1 )(β + n + 1 ) . n!(α + β + γ + n + 1 )(α + β + γ + 2n + 1 ) (9.3)
This is the adjoint of (8.11). Ismail and Zeng [29] established the connection relations stated in the next theorem.
We define the two variable polynomial Mm,n (z1 , z2 ) by
Theorem 8.6. We have the connection relation
Mm,n (z1 , z2 ) =
(β ,γ )
(9.4)
( m ≥ n ).
for m ≥ n and
j=0
(8.31) For m ≥ n, we have the special cases
n (−β ) j Hm,n (z1 , z2 ) = n! (−1 ) j Zm(β−)j,n− j (z1 , z2 ). j!
(8.34) (β )
(8.35)
(8.36) It is clear that the generating function (8.36) implies the identity
(β )
(γ )
Z j,k (z1 , z2 )Zm− j,n−k (z3 , z4 )
m≥ j≥k≥0
( j − k )! ( m − n − j + k )!
.
(8.37)
(α +β +1 )
(x + y ) =
n (β ) Lk(α ) (x )Ln (y ).
(−1 )n n! (β ,0) M (z, z¯ ), (β + 1 )n m,n
m ≥ n.
(9.8)
From (9.1) we get
ck (n, α ) =
(m + β + γ + 1 )n−k (γ + 1 )m (−1 )n−k , k!(n − k )!(γ + 1 )m−k
(9.9)
where α = m − n with m ≥ n. Then ,γ ) (z , z ) (β + γ + m + n + 2 )z2 Mm(β+1 ,n 1 2
(β ,γ )
(β ,γ )
= (γ + m + 1 )Mm,n (z1 , z2 ) − (n + 1 )Mm+1,n+1 (z1 , z2 ), (β ,γ ) ( z1 , z2 ) (β + γ + m + n + 1 )z1 Mm,n (β ,γ )
(β ,γ )
= (β + γ + m + 1 )Mm+1,n (z1 , z2 ) − (β + n )Mm,n−1 (z1 , z2 )
(9.10)
(9.11)
and (β ,γ ) ,γ ) ,γ ) (cn −z1 z2 )Mm,n (z1 , z2 )=an Mm(β+1 (z , z ) + bn Mm(β−1 ( z , z ), ,n+1 1 2 ,n−1 1 2
This is an analogue of the convolution identity
Ln
(9.7)
for m ≥ n as it is stated in [29]. The disk polynomials defined in [11] can be expressed as β
uvz z + z u um vn 1 2 1 (β ) Zm,n (z1 , z2 ) = (1 + uv )−β −1 exp . ( m − n )! 1 + uv m≥n≥0
( z1 + z3 , z2 + z4 ) =
(9.6)
(γ + m + 1 ) (β + n + 1 ) δm,p δn,q , n!(β + γ + m + 1 )(β + γ + m + n + 1 )
Pm,n (z ) =
∂ Zm,n (β ) + nz1 Zm,n = 0. ∂ z2
,γ ) (z1 , z2 ) = Mm(β+1 ( z , z ). ,n 1 2
(β ,γ ) (β ,γ ) dθ 2γ 2 β |z|≤1 Mm,n (z, z¯ )M p,q (z, z¯ )r (1 − r ) rdr π
=
Ismail and Zeng [29] gave the generating function
(β +γ +1 )
(9.5)
( β ,γ )
(8.33)
Therefore Theorem 6.4 shows that
Zm,n
z1 Mm,n
The Laguerre differential equation is [45,50],
xy + (1 + α − x )y + ny = 0.
∂
m < n.
Clearly, Mm,n (z1 , z2 ) satisfy the orthogonality relation
j=0
+ ( 1 + β + m − n − z1 z2 )
(β ,γ )
(β ,γ +1 )
(8.32)
j=0
m,n z22
(β ,γ )
Mm,n (z1 , z2 ) = Mn,m (z2 , z1 ), Then,
n n (β ) n!Zm,n (z1 , z2 ) = (β ) j (−1 ) j Hm− j,n− j (z1 , z2 ), j
z2
n (m + β + γ + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k k!(n − k )!(γ + 1 )m−k k=0
n (β − γ ) j Zm,n (z1 , z2 ) = (−1 ) j Zm(γ−)j,n− j (z1 , z2 ) j!
(β )
∂ 2 Z (β )
β ,γ
(9.12) (8.38)
k=0
Problem. Al-Salam and Chihara characterized all 1-D orthogonal polynomials satisfying formulas of convolution type in [2]. They discovered the Al-Salam–Chihara polynomials through this characterization, [25]. It will be interesting to solve the corresponding 2D characterization problem.
where
an =
(n + 1 )(β + γ + m + 1 ) , (β + γ + m + n + 1 )(β + γ + m + n + 2 )
(9.13)
bn =
(γ + m )(β + n ) , (β + γ + m + n )(β + γ + m + n + 1 )
(9.14)
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n (γ + m ) (n + 1 )(γ + m + 1 ) − . β +γ +m+n+2 β +γ +m+n
cn =
=n
(9.15)
k=0
It must be noted that (9.12) is essentially the three term recurrence relation for Jacobi polynomials [50]. It is clear that
δ φn (x, α ) + Bn δx φn (x, α ) + Cn φn (x, α ) = 0,
An x2
n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k − (k − 1 )!(n − k )!(γ + 1 )m−k k=1
(β ,γ )
An = 1 − x,
(9.17)
α + γ − x(α + β + γ + 1 ),
(9.18)
Similarly, Ismail and Zeng [29] established the following generating functions: ∞
(β ,γ )
Mm,n (z1 , z2 )um vn
m,n=0
Cn = xn(α + β + γ + n + 1 ).
(9.19)
Then for m ≥ n, Mm,n (z1 , z2 ) satisfy the following second order partial differential equation
(1 − z1 z2 )δz22 f (z1 , z2 ) + {m − n + γ − z1 z2 (β + γ + m − n + 1 )}δz2 f (z1 , z2 ) + z1 z2 n ( β + γ + m + 1 ) f ( z1 , z2 ) = 0.
2β +γ
=
( β ,γ )
×
ρ
(9.20)
( 1 + u v + ρ ) −β ( 1 − u v + ρ ) −γ
1 1 + −1 , 1 − 2z1 u/(1 + ρ − uv ) 1 − 2vz2 /(1 + ρ − uv ) (9.23)
and
(β ,γ )
Mm,n (z1 , z2 )
An equivalent form is
( 1 − z1 z2 )z2
(β +1,γ )
= nMm,n (z1 , z2 ) − (γ + m )Mm−1,n−1 (z1 , z2 ).
(9.16)
where
Bn =
n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k k!(n − k )!(γ + 1 )m−k
m≥n≥0
∂ 2 f ( z1 , z2 ) ∂ z22
2β +γ
=
+ [1 + m − n + γ − z 1 z 2 ( 2 + γ + β + m − n ) ] +z1 n ( m + β + γ + 1 ) f ( z1 , z2 ) = 0.
ρ
∂ f ( z1 , z2 ) ∂ z2
× exp (9.21)
Similarly we establish
um vn ( m − n )!
( 1 + u v + ρ ) −β ( 1 − u v + ρ ) −γ 2z u 1
1 − uv + ρ
,
where
ρ = ( 1 − 2 u v ( 1 − 2 z 1 z 2 ) + u 2 v2 ) 1/2
∂ 2 f ( z1 , z1 ) ( 1 − z1 z2 )z1 ∂ z21
(9.24)
(9.25)
They also verified the limiting relation
+ [1 − m + n + γ − z 1 z 2 ( 2 + γ + β − m + n ) ] + z2 m ( n + β + γ + 1 ) f ( z1 , z2 ) = 0.
(β ,γ )
β →∞
(9.22)
Theorem 9.1. For m ≥ n ≥ 0, β > −1, γ > −1 we have
∂ (β ,γ ) (β +1,γ ) M (z , z ) = −(β + γ + m + 1 )Mm,n ( z1 , z2 ) ∂ z2 m,n 1 2 (β ,γ ) (β ,γ ) +1,γ ) δz2 Mm,n (z1 , z2 ) = nMm,n (z1 , z2 ) − (γ + m )Mm(β−1 (z , z ) ,n−1 1 2 (β +1,γ ) (β ,γ ) = (β + γ + m + 1 ) Mm,n (z1 , z2 ) − Mm,n ( z1 , z2 ) , = (β + γ + m + 1 )Mm,n
( z1 , z2 )
(β ,γ )
m≥n≥0
(β ,γ )
−(β + γ + n + 1 )δz2 Mm,n (z1 , z2 ) = (m − n )(β + γ + m + 1 )Mm,n
( z1 , z2 ),
(β ,γ ) (β ,γ ) (z1 , z2 ) = (m − n )Mm,n ( z1 , z2 ), (δz1 − δz2 )Mm,n (β ,γ )
(β + γ + m + 1 )δz1 Mm,n (z1 , z2 ) (β ,γ )
−(β + γ + n + 1 )δz2 Mm,n (z1 , z2 ) (β +1,γ )
= (m − n )(β + γ + m + 1 )Mm,n
2β +γ (1 + uv + ρ )−β (1 − uv + ρ )1−γ , ρ (1 − uv − 2uz1 + ρ ) (9.26)
with ρ as in (9.25). 10. q-Laguerre type 2 − D polynomials For α > −1 the q-Laguerre polynomials n q(α +n−k )(n−k ) (−x )n−k , (q; q )k (q, qα+1 ; q )n−k
(10.1)
k=0
(β ,γ ) ( z1 , z2 ) (β + γ + m + 1 )δz1 Mm,n (β +1,γ )
(β ,γ )
um vn Mm,n (z1 , z2 )=
Ln(α ) (x; q ) = (qα +1 ; q )n
−(β + γ + n + 1 )Mm,n (z1 , z2 ),
(γ ) (z1 z2 ) = (−1 )n Zm,n ( z1 , z2 ),
and found the generating relation
(β ,γ ) (β ,γ ) +1,γ ) δz1 Mm,n (z1 , z2 ) = mMm,n (z1 , z2 ) − (γ + m )Mm(β−1 (z , z ) ,n−1 1 2 (β +1,γ )
(γ +m−n )
lim Mm,n (z1 , z2 /β ) = z1m−n Ln
∂ f ( z1 , z2 ) ∂ z2
( z1 , z2 ).
and (β ,γ ) δz2 Mm,n ( z1 , z2 ) n (β + γ + m + 1 )n−k (γ + 1 )m z1m−k (−z2 )n−k (n − k ) = k!(n − k )!(γ + 1 )m−k k=0
are orthogonal to infinitely many measures. Some orthogonality measures are
0
∞
(α ) Lm (x; q )Ln(α ) (x; q )xα dμ(x|q ) = ζn (α )δm,n ,
(10.2)
are given by
d μ ( x|q ) =
xα d x
(−x; q )∞
,
(q−α ; q )∞ (qα+1 ; q )n , ( q; q )∞ qn ( q; q )n ∞ xδ (x − cqk ) d μ ( x|q ) = , (−x; q )∞ k=−∞
ζn (α ) = (−α )(α + 1 )
(10.3)
−α
(q, −cqα+1 , − q c ; q )∞ (qα+1 ; q )n ζn (α ) = α+1 (q , −c, − qc ; q )∞ c−α−1 (q; q )n qn
(10.4)
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and
d μ ( x|q ) =
ζn (α ) =
x
−c
−λx, − λqx ; q
dx
∞ , −x, − qc x, − λqqc x ; q ∞ ∞ xα −c − x, − q ; q dx λx ∞ , −x, − qc x, − λqqc x ; q 0 ∞
λ
zm,n (z1 , z2 |q ) =
(β +m−n )
z1m−n Ln
1 − qβ
= (10.5)
where c, λ > 0. Following the procedure outline in Sections 6 we let (β )
(z1 z2 ; q ), m ≥ n,
(β )
zn,m (z2 , z1 |q ),
(10.6)
m < n,
1−q
(β )
zm,n (z1 , z2 |q ) =
k=0
(−1 )
wβ ( z1 z2 ; q )
n−k
( q; q )k ( q; q )
m−k n−k z2 m z1 . β +1 ; q n−k q m−k
(
)
(10.7)
(β )
z1 zm,n (z1 , z2 |β ) = zm+1,n (z1 , z2 |β ), (0 ) zm,n ( z1 , z2 |q ) =
=
(β )
Dq,z2 z2m−n wβ (z1 z2 ; q )zm,n (z1 , z2 )
1 − qn+1 (β ) z (z1 , z2 ), (10.18) 1 − q m,n+1
=
( z1 z2 ; q )
(−1 )n h ( z , z |q ), (q; q )n m,n 1 2
m ≥ n.
where m > n.
(10.8)
(β )
+
(10.9)
for m, n, s, t = 0, 1, . . . and dμ(x|q), ζ n (α ) may be any pair from (10.3), (10.4) or (10.5).
where θq,z = zDq,z has a solution y(x ) = Ln(α ) (x; q ). This leads to:
2 (1 + qz1 z2 )θq,z f+ 2
+
In the case
(10.10)
+
Theorem 10.2. For β > −1 and m ≥ n we have (β )
q(m+1+β ) z2 zm+1,n (z1 , z2 ) = (qn+1 − 1 )zm+1,n+1 (z1 , z2 ) (10.11)
(β )
qn z1 zm,n+1 (z1 , z2 ) = zm+1,n (z1 , z2 ) − zm,n−1 (z1 , z2 ) 1 + q ( 1 − qn − qm+β ) − z1 z2 q = (1 − q
1+n
(10.12)
pn ( x; qα |q ) =
(k−n )(n+k−1 ) n 2 ( q; q )n q (−x )n−k α +1 (q; q )k (q, q ; q )n−k
(β )
(β )
zm,n (z1 , z2 )
(β )
(β ) )zm+1,n+1 (z1 , z2 ) + q(1 − qβ +m )zm ( z , z ). −1,n−1 1 2
(β )
wm,n (z1 , z2 |q ) =
z1m−n pn (z1 z2 ; qβ +m−n |q ), m ≥ n, (β )
wn,m (z2 , z1 |q ),
for β > −1 and m, n = 0, 1, . . .. Then for m ≥ n find that (β +1 )
(β )
z1 wm,n (z1 , z2 |β ) = wm+1,n (z1 , z2 |β ),
Theorem 10.3. For β > −1 let wβ (x; q ) = xβ /(−x; q )∞ . Then we have
wm,n (z1 , z2 |q ) =
(β )
z1n−m (β )
Dq,z2 zm,n (z1 , z2 ) =
(β )
zm,n (z1 , z2 ) =
=
(1 − q )m+n Dnq,z1
β z Dm 2
q,z2
−β z2 wβ
(−1 ) (q; q )n wβ (z1 z2 ) m
(10.14)
(10.15)
( z1 z2 )
(10.25)
n (q; q )n (qβ +1 ; q )m−n z1m−k z2n−k q−(2 )−(2 ) k
(β )
k=0
n
(−1 )n−k (q; q )k (q; q )n−k (qβ +1 ; q )m−k
. (10.26)
qβ z2 ( β ) z (qz1 , z2 ), q − 1 z1 m,n−1
qβ ( β ) z (qz1 , z2 ), q − 1 m,n−1
(10.24)
m < n,
The raising and lowering relations of the q-Laguerre polynomials lead to the following q-difference relations,
Dq,z1 z1n−m zm,n (z1 , z2 )
(10.23)
k=0
we define the 2D polynomials wm,n (z1 , z2 |q ) through
β +m+n+1
(10.13)
(10.22)
q ( 1 − qβ +m )z1 z2 f = 0. ( 1 − q )2
Starting with the little q-Laguerre [38] or Wall polynomials
(β )
and
1 − qβ +m−n + (2 − qβ +m+1 )qz1 z2 θq,z1 f 1−q
[9]
+ (1 − qm+β +1 )zm,n (z1 , z2 ), (β )
(10.21)
qz1 z2 (1 − qn ) f =0 ( 1 − q )2
2 (1 + qz1 z2 )θq,z f− 1
Applying formulas (6.7) and (6.8) lead to the following theorem.
(β )
qn−m−β − 1 − (2 − qn )qz1 z2 θq,z2 f 1−q
and
(qα+1 ; q )n q(α+n−k)(n−k) (−1 )n−k (q; q )k (q, qα+1 ; q )n−k
(β )
(10.20)
qx(1 − qn ) y ( x ) = 0. ( 1 − q )2
(β )
= ζm∧n (|m − n| + β )δm,s δn,t ,
ck (n, α ) =
1 − qα − 2qα +1 x + qα +n+1 x θq,x y(x ) qα ( 1 − q )
Theorem 10.4. For m ≥ n the function f = zm,n (z1 , z2 ) satisfies the second order q-difference equations
(β )
zm,n (z, z¯|q )zs,t (z, z¯|q )r 2β dμ(r 2 |q )dθ
1 − qn+1 z1 (β ) z ( z1 , z2 ), 1 − q z2 m,n+1 (10.19)
2 (1 + qx )θq,x y (x ) +
Theorem 10.1. Let β > −1. Then the following orthogonality relations
(10.17)
The following second order q-difference equation
(β +1 )
R2
(β )
z2m−n wβ q(β +m−k )(n−k ) (qβ +1 ; q )
qβ z2 ( β ) z ( z1 , z2 ), 1 − q m+1,n
Dq,z1 wβ (z1 z2 ; q )zm,n (z1 , z2 )
Therefore
(β )
zm,n (z1 , z2 ) −
where m ≥ n and
and use the notation n
wβ ( z1 z2 )
λ λ
(β )
z2 Dq,z2 wβ (z1 z2 )zm,n (z1 , z2 )
17
The orthogonality relation for the little q-Laguerre polynomials is ∞
q(α +1)k (qk+1 ; q )∞ pm (qk ; qα |q ) pn (qk ; qα |q )
k=0
,
(10.16)
(q; q )∞ q(α+1)n (q; q )n δm,n = , (qα+1 ; q )∞ (qα+1 ; q )n
(10.27) m, n = 0, 1, . . . .
This leads to the orthogonality relation below.
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Theorem 10.5. For m, n, s, t = 0, 1, . . . and β > −1 we have the following orthogonality relations
(β )
(β )
wm,n (z, z¯|q )ws,t (z, z¯|q )r 2β dμ(r 2 |q )dθ
R2
(β )
=
β +m−n )w ( z , z |q ) z1 ( 1 − q m,n+1 1 2 z2 qβ +m−n−1 (1 − q )
and
= ζm∧n (|m − n| + β )δm,s δn,t ,
(10.28)
(β )
z2 Dq−1 ,z2 w(z1 z2 ; β|q )wm,n (z1 , z2 |q )
where z = reiθ ,
d μ ( r |q ) =
∞
(β )
x(xq; q )∞ δ (x − qk ), (10.29)
The function y(x ) = pn (x; qα |q ) satisfies the q-difference equation 2 qα +n−1 θq,z y (z ) +
q
n−1
+ qα +n−1 − z 1−q
qβ +1−n 1 − qβ +m+1 1 − qβ ( β ) (β ) wm,n (z1 , z2 |q )− z2 wm+1,n (z1 , z2 |q ). 1−q 1 − q 1−qβ +m−n+1 (10.39)
=
k=0
θq,z y(z ) +
z (1 − q ) y (z ) = 0 ( 1 − q )2 (10.30) n
Applying the procedure of Section 6 to (k−n )(n+k−1 )
ck (n, α ) =
(β )
Theorem 10.6. For β > −1, m ≥ n the polynomial wm,n (z1 , z2 |q ), satisfies the q-partial difference equations
qn−1 − qβ +m−1 − z1 z2 ( 1 − qn )z1 z2 θq,z2 f + f = 0, q,z2 f + 1−q ( 1 − q )2 (10.31) qn − qβ +m + qz1 z2 q ( 1 − qβ +m )z1 z2 θq,z1 f + f = 0. 1−q ( 1 − q )2 (10.32)
Theorem 10.8. For β > −1, m ≥ n we have (β )
qn (1 − qm−n+β +1 )
z1 ( 1 − q
The following theorem follows from the raising and lowering relations of the little q-Laguerre polynomials. Theorem 10.7. For β > −1 and m ≥ n we have (β )
Dq,z1 z1n−m wm,n (z1 , z2 |q )
z1n−m
(β )
=−
1−n n z2 q (1 − q )wm,n−1 (z1 , z2 |q ) , z1 (1 − q )(1 − qβ +m−n+1 )
(β )
Dq,z2 wm,n (z1 , z2 |q ) = −
(β )
q1−n (1 − qn )wm,n−1 (z1 , z2 |q )
(1 − q )(1 − qβ +m−n+1 )
,
(10.35)
= (1 − q −q
m−n+β +1
qm(n−1)+n(β −1)−(2 ) (1 − q )m+n (−1 )m (qβ +m−n+1 ; q )n
−β × Dnq−1 ,z1 (z1 z2 )β Dm w(z1 z2 ; β|q ) q−1 ,z2 (z1 z2 )
. (10.36)
For m > n we have
(β )
Dq−1 ,z1 w(z1 z2 ; β|q )wm,n (z1 , z2 |q )
Dq−1 ,z2
(β )
(z1 z2 ; β|q )wm,n (z1 , z2 |q )
z2m−n w
(1 − q )wm,n−1 (z1 , z2 |q ),
(β )
qn + qm+β (1 − qn − qn+1 ) − z1 z2 wm,n (z1 , z2 |q ) (β )
= qn (1 − qm+β +1 )wm+1,n+1 (z1 , z2 |q ) m+β
(10.43)
(β )
(1 − q )wm−1,n−1 (z1 , z2 |q ). n
( α,β )
11. The polynomials {Mn
( z1 , z2 |q )}
For α , γ > −1, the little q-Jacobi polynomials k n ( q; q )n (qα+γ +n+1 ; q )n−k q(2 ) (−x )n−k n (q; q )k (q, qα+1 ; q )n−k q (2 )
k=0
(11.1) satisfy the orthogonality relation (6.1) with
d μ ( x; γ |q ) =
∞ (qx; q )∞ x δ ( x − qk ), (qγ +1 x; q )∞ k=0
(q, qα+γ +n+1 ; q )∞ (q; q )n qn(α+1) , α +1 ; q ) (1 − qα +γ +2n+1 ) ∞ (q n
(qα+1 , qγ +n+1 ; q )
(11.2) [38]. Following our general construction in Section 6, we define the polynomials
(β ,γ )
Mm,n (z1 , z2 |q ) =
z1m−n pn (z1 z2 , qβ +m−n , qγ |q ), m ≥ n, (β ,γ )
Mn,m (z2 , z1 |q ),
m < n.
(11.3)
(β ,γ )
Mm,n (z1 , z2 |q )
(β ) (1 − qβ +m−n )wm,n ( z , z |q ) +1 1 2 = , qβ +m−n−1 (1 − q )
z2m−n w
(10.42)
(β )
n
In other words
w(z1 z2 ; β|q )
(β ) )wm ( z , z |q ) +1,n 1 2
and
n
(β )
w(z1 z2 ; β|q )wm,n (z1 , z2 |q ) =
)wm,n (z1 , z2 |q )
m+β +1
ζn (α , γ |q ) =
and
(β )
(β )
pn ( x; qα , qγ |q ) =
(10.34)
m−n+β +1
+q (10.33)
(β )
= wm,n (z1 , z2 |q ) − wm+1,n+1 (z1 , z2 |q ), (10.41)
Let
xα .
(10.40)
we can derive the recurrences in the following theorem.
qβ +m−1 θ 2
w(x; α ) = (qx; q )∞
2 ( q; q )n q (−1 )n−k (q; q )k (q, qα+1 ; q )n−k
z2 wm+1,n (z1 , z2 |q )
where θq,z = zDq,z . This gives the following theorem.
wm,n (z1 , z2 |q )
(q; q ) q(α+1)n (q; q )n ζn (α ) = α+1∞ . (q ; q )∞ (qα+1 ; q )n
2 qn θq,z f− 1
(10.38)
(10.37)
(z1 z2 ; β|q )
=
k n (qβ +γ +1 ; q )m+n−k q(2 ) z1m−k (−z2 )n−k (q; q )n (qβ +1 ; q )m−n , n (q; q )k (q; q )n−k (qβ +1 ; q )m−k (qβ +γ +1 ; q )m q(2 ) k=0
(11.4) (β +1,γ )
z1 Mm,n
,γ ) (z1 , z2 |q ) = Mm(β+1 ( z , z |q ) ,n 1 2
(11.5)
for m ≥ n. Now Theorem 6.1 leads to:
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Theorem 11.1. For m, n, s, t = 0, 1, . . . and β , γ > −1 we have the following orthogonality relation
(β ,γ )
R2
(β ,γ )
Mm,n (z, z¯|q )Ms,t
(z, z¯|q )r2β dμ(r2 ; γ |q )dθ
(11.6)
= ζm∧n (|m − n| + β , γ |q )δm,s δn,t , where dμ(x; γ |q), ζ n (α , γ |q) are given in (11.2).
ck (n, α ) =
( q; q )n
( q; q )k
(q, qα+1 ; q )
(−1 ) . (n2 )−(2k ) n−k q n−k
n−k
(β ,γ )
(β ,γ )
q−α − 1 − q1−n−α x − qγ +2 (2 − qn )x 2 (1 − qγ +2 x )θq,x y (x ) + θq,x y(x ) 1−q 1 + q1−α x − (q1−n−α + qγ +n+2 )x − y ( x ) = 0. ( 1 − q )2
qn−m−β − 1 − q1−m−β z1 z2 − qγ +2 (2 − qn )z1 z2 (1 − q )(1 − qγ +2 z1 z2 )
θq,z2 f
1 + (qn+1−m−β − q1−m−β − qγ +n+2 )z1 z2 − f =0 (1 − q )2 (1 − qγ +2 z1 z2 )
+
f+
θq,z1 f
(1 − q )2 (1 − qγ +2 z1 z2 )
f = 0.
When β , γ > −1 the raising and powering operators for the polynomials {pn (x; qα , qγ |q)} with
(qx; q )∞ xα . (qγ +1 x; q )∞
(11.13)
lead to the following ladder operators (β ,γ )
Mm,n (z1 , z2 |q ) − Mm,n (z1 , qz2 |q ) =−
q1−n (1 − qn )(1 − qm+β +γ −1 ) (β ,γ +1 ) z2 Mm,n−1 (z1 , z2 |q ), 1 − qm−n+β
where m ≥ n, and
(β ,γ )
Dq−1 ,z1 w(z1 z2 ; qβ , qγ |q )Mm,n (z1 , z2 |q )
f ( z1 , z2 ) =
(12.1)
∞
a j,k z1j z2k
(12.2)
j,k=0
if and only if
f ( z1 , z2 ) =
∞
a j,o
j=1
+
n
n! z1j (β + j ) L ( z1 z2 ) ( β + j + 1 )n n
a0,k z2k 2 F1
−n + k, 1 z z . k + 1, β + 1 1 2
(12.3)
Proof. Substitute the power series (12.2) for f in (8.25) and equate coefficients of like powers of z1 and z2 to find that
a j,k =
k−1−n a , k(β + j ) j−1,k−1
j, k > 0.
We iterate this and find that
⎧ ⎪ ⎨
(−n )k a j−k,0 , j≥k k! ( β + j − k + 1 )k = ⎪ ⎩ (k − j − n ) j (k − j )! a0,k− j , j ≤ k, k! ( β + 1 ) j
(11.12)
(β ,γ )
∂∂ z2 f = −n f,
Theorem 12.2. In order for the equation
q1−n − q1+m−n+β + qγ +2 (qm+β + q2(m−n+β ) − 1 ) z1 z2 − q2(m−n+β )
w ( x; qα , qγ |q ) =
z1
This proves the theorem.
(1 − q )(1 − qγ +2 z1 z2 )
has a power series solution
a j,k (11.11)
β − z1 z2
k=0
Theorem 11.2. For m ≥ n and β , γ > −1, the function f = ( β ,γ ) Mm,n (z1 , z2 |q ) satisfies
z1 z2 2qγ +2 − q1−n − qm+β +γ +2 + qm−n+β − 1
∂∂ z1 ∂∂ z2 f +
(11.10)
where y(x ) = pn (x; qα , qγ |q ), θq,z = zDq,z .
θ
where m ≥ n + 1.
Theorem 12.1. The partial differential equation (8.25), namely,
that the polynomials {pn (x; qα , qγ |q)} satisfy the following second order difference equation
2 q,z1
z1 (1 − qm−n+β )w(z1 z2 ; qβ , qγ −1 |q ) (β ,γ −1) Mm,n+1 (z1 , z2 |q ), (11.16) z2 qm−n+β −1 (1 − q )
(11.8)
(1 − qβ +m+1 )(1 − qβ +γ +m+1 ) (β ,γ ) = M ( z1 , z2 |q ) (1 − qβ +m−n+1 )(1 − qβ +γ +m+n+1 ) m+1,n qm−n+β +1 (1 − qn )(1 − qγ +n ) (β ,γ ) + M (z1 , z2 |q ). (11.9) (1 − qβ +m−n+1 )(1 − qβ +γ +m+n+1 ) m,n−1 The above formulas hold for β , γ > −1 and m ≥ n. Next we note
and
In this section we study polynomial solutions to partial differential equations. We are looking for polynomial solutions to the second order partial differential equation (8.25). Let f =
∞ j k m,n=0 z1 z2 and substitute in (8.25).
(β ,γ )
2 θq,z f+ 2
(β ,γ )
12. Polynomial solutions to differential equations
z1 Mm,n (z1 , z2 |q )
(11.15)
(11.7)
(1 − qβ +γ +m+n+2 ) (β ,γ ) z2 Mm+1,n (z1 , z2 |q ) qn (1 − qβ +m−n+1 ) = −Mm+1,n+1 (z1 , z2 |q ) + Mm,n (z1 , z2 |q ),
(1 − qm−n+β )w(z1 z2 ; qβ , qγ −1 |q ) (β ,γ −1) Mm,n+1 (z1 , z2 |q ), qm−n+β (1 − q )
z2n−m Dq−1 ,z2 z2m−n w(z1 z2 ; qβ , qγ |q )Mm,n (z1 , z2 |q ) =
Following the construction of §6 we derive the recurrence below via the choice
(qα+γ +n+1 ; q )
=
19
(11.14)
∂z1 ∂z2 f +
β z1
− z2
∂z2 f = λ f
(12.4)
to have a polynomials solution in z2 , it is necessary and sufficient that λ = −n, n = 0, 1, 2, . . . , in which case the function f will be given as in Theorem 12.1. References [1] S.T. Ali, F. Bagarello, G. Honnouvo, Modular structures on trace class operators and applications to Landau levels, J. Phys. A 43 (2010) 17. 105202. [2] W.A. Al-Salam, T.S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976) 16–28. [3] G.E. Andrews, R.A. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. [4] P. Appell, J.K. de Fériet, Fonctions Hypergéométrique et Hypersphérique, Polynomes D’hermite, Gauthier–Villars, Paris, 1926. [5] R.A. Askey, A Note on the History of Series, in: Technical Report 1532, Mathematics Research Center, University of Wisconsin. [6] R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Academic Press, 1964. [7] J.L. Burchnal, A note on the polynomials of Hermite, Q. J. Math. 12 (1941) 9–11.
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Please cite this article as: M.E.H. Ismail, R. Zhang, A review of multivariate orthogonal polynomials, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2016.11.001