Structure relations for the bivariate big q-Jacobi polynomials

Applied Mathematics and Computation 219 (2013) 8790–8802

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Structure relations for the bivariate big q-Jacobi polynomials Stanisław Lewanowicz ⇑, Paweł Woz´ ny, Rafał Nowak Institute of Computer Science, University of Wrocław, ul. F. Joliot-Curie 15, 50-383 Wrocław, Poland

a r t i c l e

i n f o

Keywords: Bivariate big q-Jacobi polynomial Structure relation

a b s t r a c t We give structure relations for the orthogonal polynomials in two variables, defined by Lewanowicz and Woz´ ny [S. Lewanowicz, P. Woz´ ny, J. Comput. Appl. Math. 233 (2010) 1554–1561]

P n;k ðx; y; a; b; c; d; qÞ :¼ Pnk ðy; a; bcq

2kþ1

k

; dq ; qÞyk ðdq=y; qÞk P k ðx=y; c; b; d=y; qÞðn

P 0; k ¼ 0; 1; . . . ; nÞ where q 2 ð0; 1Þ; 0 < aq; bq; cq < 1; d < 0, and P m ðt; a; b; c; qÞ are univariate big q-Jacobi polynomials. We discuss in full detail the case of the polynomials Pn;k ðx; y; a; b; c; 0; qÞ, which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, SIAM J. Algebra. Discr. Methods 1 (1980) 137–151]. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction One of the many characteristic properties of the classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel polynomials) is the existence of the structure relation [1] (see also [19])

/ðxÞp0n ðxÞ ¼ an pnþ1 ðxÞ þ bn pn ðxÞ þ cn pn1 ðxÞ;

ð1:1Þ

where / is a fixed polynomial of degree at most 2. Maroni [20,21] introduced semi-classical orthogonal polynomials, satisfying the more general structure relation

/ðxÞp0n ðxÞ ¼

nþt X

anj pj ðxÞ;

j¼ns

where / is a polynomial, and s; t are independent of n. Later on, it was shown that relation (1.1) also characterizes discrete classical orthogonal polynomials (Hahn, Krawtchouk, Meixner, and Charlier polynomials) [12] as well as q-classical orthogonal polynomials of the q-Hahn class [22], with the derivative being replaced by the difference operator Df ðxÞ ¼ f ðx þ 1Þ  f ðxÞ and the q-derivative operator Dq f ðxÞ ¼ ðf ðqxÞ  f ðxÞÞ=ððq  1ÞxÞ, respectively. Recently, Koornwinder [17] gave an explicit structure relation for Askey–Wilson polynomials. For bivariate orthogonal polynomials, structure relations were studied in [9–11] (classical case) and [2,3,8] (semi-classical case). Structure relations for bivariate discrete classical orthogonal polynomials on the uniform lattice are obtained in [24]; the case of bivariate q-classical orthogonal polynomials was studied by Rodal [23] (see our discussion in Remark 1). The aim of the present paper is to go further in the study of the bivariate big q-Jacobi polynomials giving structure relations for them. These polynomials were introduced by the first two authors in [18] by ⇑ Corresponding author. E-mail addresses: [email protected] (S. Lewanowicz), [email protected] (P. Woz´ ny), [email protected] (R. Nowak). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.02.059

S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802 2kþ1

Pn;k ðx; y; a; b; c; d; qÞ :¼ Pnk ðy; a; bcq

k

; dq ; qÞyk ðdq=y; qÞk P k ðx=y; c; b; d=y; qÞ

ðn P 0; k ¼ 0; 1; . . . ; nÞ;

8791

ð1:2Þ

where q 2 ð0; 1Þ; 0 < aq; bq; cq < 1; d < 0, and

!  qm ; abqmþ1 ; t  ðm P 0Þ q; q aq; cq

Pm ðt; a; b; c; qÞ :¼ 3 /2

ð1:3Þ

are univariate big q-Jacobi polynomials (see, e.g. [4, Section 7.3], or [15, Section 14.5]). For the notation used, see Section 2. In [18], we gave some basic properties of polynomials (1.2). First, we showed that they form an orthogonal system with respect to the linear functional u defined by

hu; pi :¼

Z

aq

dq

Z

cqy

ðdq=y; x=ðcyÞ; x=d; y=a; y=d; qÞ1 pðx; yÞdq xdq y; yðd=ðcyÞ; cqy=d; x=y; bx=d; y; qÞ1

dq

where p is any two-variable polynomial. Second, we proved that the following three-term relations hold:

x Pn ¼ An;1 Pnþ1 þ Bn;1 Pn þ C n;1 Pn1 ; y Pn ¼ An;2 Pnþ1 þ Bn;2 Pn þ C n;2 Pn1 ;

 ðn P 0Þ;

ð1:4Þ

where we used the notation Pn :¼ ½Pn;0 ; Pn;1 ; . . . ; P n;n T , and An;i ; Bn;i and C n;i are tridiagonal (i ¼ 1) or diagonal (i ¼ 2) matrices of appropriate dimensions. Last, we showed that for any n P 0, the polynomial vector Pn satisfies the linear second-order partial q-difference equation 

þ

l11 Dq;x Dq1 ;x Pn þ l22 Dq;y Dq1 ;y Pn þ l12 Dq1 ;x Dq1 ;y Pn þ l12 Dq;x Dq;y Pn þ m1 Dq;x Pn þ m2 Dq;y Pn ¼ kn Pn ;

ð1:5Þ

with

l11 ðxÞ :¼ ðx  dqÞðx  acq2 Þ;

l22 ðyÞ :¼ ðy  aqÞðy  dqÞ;



þ

l12 ðx; yÞ :¼ q1 ðx  dqÞðy  aqÞ; l12 ðx; yÞ :¼ acq3 ðbx  dÞðy  1Þ; n o 3 m1 ðxÞ :¼ ðabcq  1Þðx  1Þ  ðacq2  1Þðdq  1Þ =ðq  1Þ; n o 3 m2 ðyÞ :¼ ðabcq  1Þðy  1Þ  ðaq  1Þðdq  1Þ =ðq  1Þ; nþ2

kn :¼ ½nq q1n ðabcq

 1Þ=ðq  1Þ;

where Dq1 ;x and Dq1 ;y are partial q-derivative operators (see Section 2), and we let

Dq1 ;z Pn :¼ ½Dq1 ;z Pn;0 ; Dq1 ;z Pn;1 ; . . . ; Dq1 ;z Pn;n T

ðz ¼ x; yÞ:

The main result of the present paper gives structure relations in the form

r1 Dq1 ;x Pn ¼ F n;1 Pnþ1 þ Gn;1 Pn þ Hn;1 Pn1 ; r2 Dq1 ;y Pn ¼ F n;2 Pnþ1 þ Gn;2 Pn þ Hn;2 Pn1 ;    where r i are certain polynomials of total degree 2, and F n;i ; Gn;i and Hn;i (i ¼ 1; 2) are tridiagonal matrices of appropriate dimensions. A general result is given in Section 3, while in Section 4 we discuss in detail the case of polynomials (1.2) with d ¼ 0, which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [6].

2. Preliminaries The hypergeometric series is defined by (see, e.g. [4, Section 2.1])

 2F1

  1 X a; b  ðaÞk ðbÞk k z :¼ z;  k!ðcÞ c k¼0

k

where a; b; c; z 2 C, and ðaÞk is the shifted factorial symbol, defined for any a 2 C by

ðaÞ0 :¼ 1;

ðaÞk :¼ aða þ 1Þ    ða þ k  1Þ

The q-shifted factorial is defined for any a 2 C by

ða; qÞk :¼

k1 Y ð1  a qj Þ ðk P 0Þ: j¼0

Assuming that 0 < q < 1, we also put

ða; qÞ1 :¼

1 Y j¼0

ð1  a qj Þ:

ðk P 1Þ:

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In the sequel, we make use of the convention

ða1 ; a2 ; . . . ; am ; qÞk :¼

m Y 



aj ; q

k

ðk ¼ 0; 1; . . . or 1Þ:

j¼1

The q-binomial coefficient is given by

 n k

:¼

q

ðq; qÞn : ðq; qÞk ðq; qÞnk

For a 2 C, we define the q-number ½aq by

½aq :¼

qa  1 : q1

The q-derivative operator is defined for q 2 C n f1g by

Dq f ðxÞ :¼

f ðqxÞ  f ðxÞ ; ðq  1Þx

while the partial q-derivative operators are defined by

Dq;x f ðx; yÞ :¼

f ðqx; yÞ  f ðx; yÞ ; ðq  1Þx

Dq;y f ðx; yÞ :¼

f ðx; qyÞ  f ðx; yÞ : ðq  1Þy

The q-integral is defined by

Z

b

f ðxÞ dq x :¼

a

Z

b

f ðxÞdq x  0

Z

a

f ðxÞdq x; 0

where

Z

a

1 X f ðxÞdq x :¼ að1  qÞ f ðaqk Þ qk :

0

k¼0

The basic hypergeometric series is defined by (see, e.g. [4, Section 10.9])

 r /s

  1 X a1 ; . . . ; ar  ða1 ; . . . ; ar ; qÞk q; z :¼  ðq; b1 ; . . . ; bs ; qÞ b ;...;b 1

s

k¼0

  k

k

ð1Þ q

!1þsr

2

zk ;

k

where r; s 2 Zþ and a1 ; . . . ; ar ; b1 ; . . . ; bs ; z 2 C. We denote the set of all polynomials in one independent variable x by P1 and the set of all polynomials in two independent variables x and y by P2 . The q-gradient (rq ) and the q-divergence (divq ) operators are defined by

rq p :¼



Dq;x p Dq;y p

;

divq

 p :¼ Dq;x p þ Dq;y q ðp; q 2 P2 Þ: q

For i ¼ 1; 2, a linear mapping u : Pi ! C is called a moment functional. For a one-variable moment functional u and r 2 P1 , we let Dq u, the q-derivative of u, and ru, multiplication of u by a polynomial, be those moment functionals defined by

hDq u; pi :¼ hu; Dq pi;

hru; pi :¼ hu; rpi ðp 2 P1 Þ:

In the two-variable case, we define the first partial q-derivatives of u, as moment functionals, by

hDq;z u; pi :¼ hu; Dq;z pi ðp 2 P2 ; z ¼ x; yÞ and we define multiplication ru, where r 2 P2 , by hru; pi :¼ hu; rpi (p 2 P2 ). Let A ¼ ½aij  and B ¼ ½bij  be polynomial matrices of dimension m  n and m  r, respectively. We define (see, e.g. [3,8,10])  the action of u over A by hu; Ai ¼ C, where C ¼ ½hu; aij i is the real matrix of dimension m  n;  the left multiplication of u by A as hAu; Bi ¼ hu; AT Bi. The distributional q-divergence operator is defined in the following way:

  Dq;x p hdivq ðMuÞ; pi ¼ hMu; rq pi ¼  u; M T ðp 2 P2 Þ; Dq;y p where M is a polynomial matrix of dimension 2  n.

S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

8793

3. The general form of the structure relations Let u be the moment functional defined by

hu; pi :¼

Z

aq

dq

Z

cqy

Wðx; yÞpðx; yÞdq xdq y ðp 2 P2 Þ;

ð3:1Þ

dq

where

Wðx; yÞ  Wðx; y; a; b; c; d; qÞ :¼

ðdq=y; x=ðcyÞ; x=d; y=a; y=d; qÞ1 : yðd=ðcyÞ; cqy=d; x=y; bx=d; y; qÞ1

ð3:2Þ

Remember that polynomials Pn;k ðx; yÞ ¼ Pn;k ðx; y; a; b; c; d; qÞ, defined by (1.2), are orthogonal with respect to this functional: hu; Pn;k Pml i ¼ Kn;k dnm dk;l for some constants Kn;k > 0 [18]. We need the following properties of the weight (3.2). Lemma 3.1. The weight function W satisfies the system of equations

Dq;x Dq;y





r1 W ¼ s1 W;  r2 W ¼ s2 W

ð3:3Þ



ð3:4Þ

or, in the equivalent form,

Dq1 ;x Dq1 ;y





rþ1 W ¼ q s1 W;  rþ2 W ¼ q s2 W;

ð3:5Þ



ð3:6Þ

where the bivariate polynomials

ri of total degree two, and si of total degree one, are given by

r1 ðx; yÞ :¼ ðx  dqÞðx  cqyÞ; rþ1 ðx; yÞ :¼ cq2 ðx  yÞðbx  dÞ; rþ ðx; yÞ  r1 ðx; yÞ s1 ðx; yÞ :¼ 1 ; ðq  1Þx

r2 ðx; yÞ :¼ ðx  yÞðy  aqÞ; rþ2 ðx; yÞ :¼ aqðy  1Þðx  cqyÞ; rþ ðx; yÞ  r2 ðx; yÞ s2 ðx; yÞ :¼ 2 : ðq  1Þy

Proof. The general theory of bivariate q-hypergeometric orthogonal polynomials developed by Rodal [23, Section 5.5] yields equations

r1 ðqx; yÞ Wðqx; yÞ ¼ rþ1 ðx; yÞ Wðx; yÞ; r2 ðx; qyÞ Wðx; qyÞ ¼ rþ2 ðx; yÞ Wðx; yÞ [23, Eqs. (5.5.39) and (5.5.40)], which are equivalent to (3.3) and (3.4), respectively. Those two equations, together with identities









r1 W ¼ q1 Dq1 ;x rþ1 W ;      Dq;y r2 W ¼ q1 Dq1 ;y rþ2 W Dq;x

[23, Eq. (5.5.49)], imply the remaining Eqs. (3.5) and (3.6). h Using the above results, we obtain the following properties of the functional (3.1). Lemma 3.2. The functional u defined by (3.1) satisfies the matrix q-Pearson equation

divq ðUþ uÞ ¼ WT u;

ð3:7Þ

or, equivalently,

divq1 ðU uÞ ¼ qWT u;

ð3:8Þ

where



U :¼

r1

0

0

r

 2



 ;

W :¼



s1 : s2

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

Proof. Using the distributional operations defined in Section 2, we obtain

#  

 

 þ  " r1 Dq;x p hDq;x rþ1 u ; pi T Dq;x p   ¼  u; ¼ ; hdivq ðUþ uÞ; pi ¼ hUþ u; rq pi ¼  u; Uþ Dq;y p rþ2 Dq;y p hDq;y rþ2 u ; pi

   s1 p hs1 u; pi ¼ : hWT u; pi ¼ h½s1 ; s2 u; pi ¼ u; hs2 u; pi s2 p Hence, Eq. (3.7) is equivalent to the system

Dq;x



Dq;y





ð3:9Þ



ð3:10Þ

rþ1 u ¼ s1 u; rþ2 u ¼ s2 u:

In a similar way, we show that Eq. (3.8) means

Dq1 ;x Dq1 ;y





ð3:11Þ



ð3:12Þ

r1 u ¼ qs1 u;



r2 u ¼ qs2 u:

We prove Eq. (3.9); the remaining equations can be proved in a similar way. We have

hDq;x





rþ1 u ; pi ¼ hu; rþ1 Dq;x pi ¼ 

Z

Z

aq

dq

cqy

Wðx; yÞ rþ1 ðx; yÞ Dq;x pðx; yÞdq xdq y:

dq

ð3:13Þ

Using the identity (see, e.g. [14, p. 74])

Z

b

Dq f ðxÞ  gðxÞdq x ¼ f ðxÞgðx=qÞjba  q1

a

Z a

b

f ðxÞ Dq1 gðxÞdq x;

we write the inner integral in (3.13) as

Z

1 Wðx=q; yÞ rþ1 ðx=q; yÞ pðx; yÞjx¼cqy x¼dq  q

cqy

pðx; yÞ Dq1 ;x

dq





rþ1 ðx; yÞ Wðx; yÞ dq x:

Now, the first term vanishes, while the second one can be, by (3.5), written as



Z

cqy

pðx; yÞ s1 ðx; yÞ Wðx; yÞdq x; dq

so that we obtain

hDq;x





rþ1 u ; pi ¼

Z

aq dq

Z

cqy

pðx; yÞ s1 ðx; yÞ Wðx; yÞdq xdq y ¼ hu; s1 pi ¼ hs1 u; pi:

dq

Hence, Eq. (3.9) follows. h Now we can prove the main result of this section. We introduce the following column vector notation:

Pn :¼ ½P n;0 ; Pn;1 ; . . . ; Pn;n T ;

ð3:14Þ

Dq1 ;z Pn :¼ ½Dq1 ;z Pn;0 ; Dq1 ;z Pn;1 ; . . . ; Dq1 ;z Pn;n T

ðz ¼ x; yÞ;

where P n;k :¼ Pn;k ðx; y; a; b; c; d; qÞ. Theorem 3.3. For n P 0, there exist unique tridiagonal matrices ðn þ 1Þ  ðn þ 2Þ; ðn þ 1Þ  ðn þ 1Þ and ðn þ 1Þ  n, respectively, of the form

2

F n;i

f0;0 6f 6 1;0 6 6 :¼ 6 6 6 4 2

Gn;i

g 0;0

6g 6 1;0 6 6 :¼ 6 6 6 4

f1;2 .. .

..

fn1;n2

fn1;n1

fn1;n

fn;n1

fn;n

.

H n;i

(i ¼ 1; 2)

of

the

size

7 7 7 7 7; 7 7 0 5

ð3:15Þ

fn;nþ1 3

g 0;1 g 1;1 .. .

and

3

f0;1 f1;1 .. .

 F n;i ; Gn;i

g 1;2 .. .

..

g n1;n2

g n1;n1

.

g n;n1

7 7 7 7 7; 7 7 g n1;n 5 g n;n

ð3:16Þ

S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

2

Hn;i

h0;0 6h 6 1;0 6 6 :¼ 6 6 6 4

8795

3

h0;1 h1;1 .. .

h1;2 .. . hn1;n2

7 7 7 7 .. ; . 7 7 7 hn1;n1 5 hn;n1

ð3:17Þ

such that the following structure relations hold:

r1 Dq1 ;x Pn ¼ F n;1 Pnþ1 þ Gn;1 Pn þ Hn;1 Pn1 ; r2 Dq1 ;y Pn ¼ F n;2 Pnþ1 þ Gn;2 Pn þ Hn;2 Pn1 ;

ð3:18Þ ð3:19Þ

where we define P1 :¼ 0. Proof. We prove the ‘‘plus’’ identity of (3.18) (the remaining identities can be proved in a similar way). Obviously, we have

rþ1 Dq;x Pn;k ¼

nþ1 X l X sþl;m Pl;m l¼0 m¼0

for some coefficients sþ l;m . A. We show that sþ ¼ 0 for l < n  1. From the orthogonality of fP n;k g with respect to u, we find that l;m

sþl;m ¼

hu; rþ1 Dq;x Pn;k  Pl;m i hu; P2l;m i

:

Using the rule (see, e.g. [14, p. 3])

Dq ðf ðxÞgðxÞÞ ¼ f ðxÞDq gðxÞ þ gðqxÞDq f ðxÞ and (3.9), we obtain

hu; P2l;m i sþl;m ¼ hu; rþ1 Dq;x Pn;k  Pl;m i ¼ ¼ ¼

ð3:20Þ

h þ1 u; Dq;x ðP n;k P l;m Þ  Pn;k Dq;x P l;m i hDq;x ð þ1 uÞ; P n;k P l;m i  hu; þ1 Pn;k Dq;x P l;m i h 1 u; Pn;k P l;m i  hu; þ1 P n;k Dq;x P l;m i

r

r

r

s

r

ð3:21Þ

with both terms vanishing when l < n  1. Here P l;m ðx; yÞ ¼ P l;m ðx=q; yÞ. B. We show that sþ ¼ 0 for m R fk  1; k; k þ 1g. First, observe that equation l;m

Wðx; y; a; b; c; d; qÞ ¼ wðx=y; c; b; d=y; qÞ v ðy; a; c; d; qÞ

ð3:22Þ

holds, where we use the notation of (3.2), and

wðx; a; b; c; qÞ :¼

ðx=a; x=c; qÞ1 ; ðx; bx=c; qÞ1

v ðy; k; l; m; qÞ :¼

ðmq=y; y=k; y=m; qÞ1 : y ðm=ðlyÞ; lqy=m; y; qÞ1

Let us recall that the univariate big q-Jacobi polynomials Pm ðt; a; b; c; qÞ (cf. (1.3)) are orthogonal with respect to the functional

hv ; pi :¼

Z

aq

wðx; a; b; cÞpðxÞdq x ðp 2 P1 Þ:

ð3:23Þ

cq

Also, we need the formula kþ1

Dq;x P n;k ðx; y; a; b; c; d; qÞ :¼

q1k ½kq ð1  bcq Þ k 2kþ1 k ; dq ; qÞPk1 ðqx=y; cq; bq; dq=y; qÞ: y ðdq=y; qÞk P nk ðy; a; bcq ð1  cqÞð1  dq=yÞ

B1. We show that sþ ¼ 0 for m > k þ 1. Using (3.22) and (3.24) in the second member of (3.20), we obtain l;m

ð3:24Þ

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

hu; P 2l;m i sþl;m ¼ hu; rþ1 Dq;x Pn;k  Pl;m i Z aq Z cqy Wðx; y; a; b; c; d; qÞ rþ1 ðx; yÞ Dq;x Pn;k ðx; y; a; b; c; d; qÞPl;m ðx; y; a; b; c; d; qÞdq xdq y ¼ dq

¼

Z

dq

kþ1

aq

v ðy; a; c; d; qÞ

dq 2kþ1

 Pnk ðy; a; bcq

q1k ½kq ð1  bcq Þ kþm y ðdq=y; qÞk ðdq=y; qÞj ð1  cqÞð1  dq=yÞ k

2jþ1

; dq ; qÞP lm ðy; a; bcq

j

; dq ; qÞ Ik;m ðyÞdq y;

with

Ik;m ðyÞ :¼

Z

cqy

wðx=y; c; b; d=y; qÞ rþ1 ðx; yÞ Pk1 ðqx=y; cq; bq; dq=y; qÞP m ðx=y; c; b; d=y; qÞdq x

dq

¼y

Z

cq

wðt; c; b; d=y; qÞ Q k ðt; yÞ Pm ðt; c; b; d=y; qÞdq t; dq=y

where Q k ðt; yÞ :¼ rþ 1 ðyt; yÞ P k1 ðqt; cq; bq; dq=y; qÞ is a polynomial in t of degree k þ 1. Notice that the last integral is the result of action of the functional (3.23) with a ¼ c; b ¼ b; c ¼ d=y on the product pðtÞ :¼ Q k ðt; yÞ P m ðt; c; b; d=y; qÞ; y being a parameter. Hence, Ik;m ðyÞ as well as sþ l;m vanishes for m > k þ 1. B2. In the remaining part of the proof, we use Eq. (3.21),

hu; P 2l;m i sþl;m ¼ hs1 u; P n;k P l;m i  hu; rþ1 Pn;k Dq;x P l;m i; to show that sþ ¼ 0 for m < k  1: l;m First, by reversing the roles of Pn;k and Pl;m in the argument of the part B1 of the proof, we easily conclude that

hu; rþ1 Pn;k Dq;x P l;m i ¼ 0 for m < k  1: Next, we obtain

hs1 u; Pn;k P l;m i ¼

Z

aq

dq

v ðy; a; c; d; qÞykþm ðdq=y; qÞk ðdq=y; qÞj Pnk ðy; a; bcq2kþ1 ; dqk ; qÞPlm ðy; a; bcq2jþ1 ; dqj ; qÞ Jk;m ðyÞdq y;

where

J k;m ðyÞ :¼ y

Z

cq

wðt; c; b; d=y; qÞ Rm ðt; yÞ P k ðt; c; b; d=y; qÞdq t;

dq=y

with Rm ðt; yÞ :¼ s1 ðyt; yÞ P m ðt=q; c; b; d=y; qÞ being a polynomial in t of degree m þ 1. Obviously, J k;m ðyÞ as well as hs1 u; Pn;k P l;m i vanishes for m < k  1. Summing up, we have shown that the following identity holds:

rþ1 Dq;x Pn;k ¼

nþ1 X kþ1 X

sþl;m P l;m ;

ð3:25Þ

l¼n1m¼k1

which is equivalent to the ‘‘plus’’ identity of (3.18) with the obvious identification

fk;m ¼ sþnþ1;m ðn; kÞ;

g k;m ¼ sþn;m ðn; kÞ;

hk;m ¼ sþn1;m ðn; kÞ:



Remark 1. Notice that in [23, Thm 5.5.2], the structure relations are given in a weaker form

arþ1 Dq;x Pn  drþ2 Dq;y Pn ¼ W n Pnþ1 þ Y n Pn þ Z n Pn1 ; ~ n Pnþ1 þ Y~ n Pn þ Z~ n Pn1 ; ar D 1 Pn  dr D 1 Pn ¼ W 1

q

;x

2

q

;y

ð3:26Þ ð3:27Þ

~ n; Y ~ n and Z ~ n are matrices of appropriate dimensions. Obviously, such identities can be easily obtained where W n ; Y n ; Z n ; W using (3.18) and (3.19).

4. Special case: bivariate little q-Jacobi polynomials Notice that

Pn;k ðx; y; a; b; c; 0; qÞ ¼ cn;k p n;k ðx; y; a; b; cj qÞ with

ð4:1Þ

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802     k nk þ þn 2 2

n nk k

cn;k :¼ ð1Þ a

c q



2kþ2 ;q ðbq; qÞk bcq nk ; ðcq; qÞk ðaq; qÞnk

where the polynomials 2kþ1

p n;k ðx; y; a; b; cj qÞ :¼ p nk ðy; bcq

; ajqÞ yk p k ðx=y; b; cjqÞ

are closely related to Dunkl’s bivariate p m ðx; a; bj qÞ :¼ pm ðx=ðbqÞ; a; bj qÞ, where mþ1

qm ; abq aq

pm ðx; a; bj qÞ :¼ 2 /1

(little)

ð4:2Þ

q-Jacobi

polynomials

[6].

Here

we

use

 !    q; qx 

the

notation

ð4:3Þ

are little q-Jacobi polynomials of one variable (see, e.g. [13, p. 182], or [15, Section 14.12]). From (4.2) and (4.3), the following power representation can be deduced:

p n;k ðx; y; a; b; cj qÞ ¼

k X ni X

bi;j ðn; kÞ xi yj ;

ð4:4Þ

i¼0 j¼ki

where





kþ1 nþkþ2 qk ; bcq ; q qkn ; abcq ;q i

iþjk : bi;j ðn; kÞ :¼ 2kþ2 ajþik ci ðq; bq; qÞi q; bcq ;q

ð4:5Þ

iþjk

The following inverse representation:

xi y j ¼

n minði;kÞ X X k¼0

ak;l ði; jÞ p k;l ðx; y; a; b; cj qÞ;

ð4:6Þ

l¼0

where n :¼ i þ j and k

   l kl

ak;l ði; jÞ :¼ ð1Þ q

þ

2



2



ð1  abcq2kþ2 Þðbq; qÞi bcqiþlþ2 ; q   nl j iþjl i i



ðaqÞ ðcqÞ : lþ1 kþlþ2 l q kl q bcq ; q abcq ;q l

ð4:7Þ

nlþ1

can be proved by induction on n, using the recurrence relations (1.4). Theorem 4.1. The following structure relations hold:

xðx  yÞDq;x p n;k ðx; y; a; b; cj qÞ ¼

nþ1 kþ1 X X

sþl;m p l;m ðx; y; a; b; cj qÞ;

ð4:8Þ

l¼n1 m¼k1 nþ1 kþ1 X X

ðy  1Þðx  cqyÞDq;y p n;k ðx; y; a; b; cj qÞ ¼

r þl;m p l;m ðx; y; a; b; cj qÞ;

ð4:9Þ

l¼n1 m¼k1

where kþ1

sþn1;k1 ¼ a2 q2n2kþ2 ½kq

ð1  cqk Þð1  bcq

2nþ1 abcq ;q

Þ

;

2

sþn1;k

¼

k

acqknþ3 ½kq

½k  1q ð1  bq Þ

½k þ 1q ð1  bq

kþ1

Þ

! 1 þ cq

 kþnþ1 2k 2kþ2 1  bcq 1  bcq 1  abcq !   2n nþkþ1 nk n  k þ 1 ð1  abcq2n Þð1  bcqnþkþ2 Þ Þ kþn 2 ð1  abcq Þð1  bcq  q ð1  bcq Þ  ½n  kq þ ; 2nþ1 2nþ2 2 2 1  abcq 1  abcq q q 

nk



kþ1

kþ1

ð1  bq Þð1  bcq Þ

2kþ1 2kþ2 2 bcq ; bcq ;q q

2

1 0 0 1 nþkþ1 nþkþ2

½2q bcq ;q bcq ;q 2n nþk 2 2 A  q bcq ; q A;  @ð1  abcq Þ@  2nþ1 2nþ2 2 1  abcq 1  abcq

sþn1;kþ1 ¼cq3k2nþ4 ½kq

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802 kþ1

sþn;k1 ¼ a2 q2n2kþ2 ½kq ðq þ 1Þ 0 sþn;k ¼ acq2 ½kq @ 0 @

ð1  cqk Þð1  bcq 2nþ1

ð1  abcq



k ½k  1q 1  bq 2k

1  bcq



nþkþ1 ½n  kq 1  bcq 2nþ1

1  abcq







kþ1 ½k þ 1q 1  bq

ð1  bq

Þ

;

1 1A þ cq

1

2kþ2

1  bcq nþkþ2 ½n  k þ 1q 1  bcq

A;

2nþ3

1  abcq

kþ1

sþn;kþ1 ¼ cq2knþ2 ½kq ½n  kq

Þ 2nþ3

Þð1  abcq

kþ1

nþkþ2

Þð1  bcq Þð1  abcq

2kþ1 2kþ2 bcq ; bcq ;q



1 0 nþkþ2 nþkþ1 bcq ; q bcq ; q Þ@ 2 2A ;  2nþ3 2nþ1 1  abcq 1  abcq

2

kþ1

sþnþ1;k1 ¼ a2 q2n2kþ3 ½kq

ð1  cqk Þð1  bcq

2nþ2 abcq ;q

Þ

;

2 kþ1

2kþ1

sþnþ1;k ¼ aqnkþ1 ½kq ð1  cqk  cqkþ1 þ bcq

sþnþ1;kþ1

Þ

ð1  bcq

nþkþ2

Þð1  bcq

2k

2kþ2

ð1  bcq Þð1  bcq

nþkþ2

Þð1  abcq Þ

; 2nþ2 Þ abcq ;q 2



kþ1 kþ1 nþkþ2 nþkþ2 ð1  bq Þð1  bcq Þ bcq ; abcq ;q 2

¼ c qkþ1 ½kq 2kþ1 2kþ2 2nþ2 bcq ; bcq ; abcq ;q 2

and

rþn1;k1 ¼ abc qnþ2 ½kq

ð1  cqk Þð1  aqnkþ1 Þ

; 2nþ1 abcq ;q 2



 2 n

bcq ð½kq  ½nq Þ aqn  qk





ðc þ 1Þ abcqnþkþ2  bcq2kþ1  1 rþn1;k ¼ 2k 2kþ2 2 nþ1 1  bcq abcq ;q 1  bcq 2

o 2 nþ2kþ2 2 2 nþkþ1 k nþ1  abc q þ acqnkþ1 ; þðb þ 1Þ cq ð1 þ qÞ  acq þ ab c3 qnþ3kþ3  abc qnþkþ3 þ abcq

r þn1;kþ1 ¼



nk 2



2

kþ1

kþ1

bc q4knþ4 ð1  q2 Þð1  bq Þð1  bcq Þð1  abcq

2kþ1 2kþ2 2nþ1 a bcq ; bcq ; abcq ;q q

kþnþ2

  Þ aqnk1 ; q 2

2

rþn;k1 ¼ abc qnþ2 ½kq



2nþ2 ð1  cqk Þ ð1  abcq Þ  ð1  aqnkþ1 Þ  ð1  aqnkþ2 Þ 2nþ1

ð1  abcq nþkþ1

rþn;k

2

¼acq

1 ½n  kq ð1  bcq þ 2nþ1 aq 1  abcq

Þ

nþkþ2



2nþ3

2k

1  bcq

½n  kq ð1  bcq

2nþ1

1  abcq

kþ1

kþ1



½k þ 1q ½n  kq ð1  bq

!

2kþ2

k

1þ

Þ

1  bcq

q½kq ð1  bq Þ 1  bcq nþkþ2

2k

kþ1



kþ1

½k þ 1q ð1  bq 2kþ2

1  bcq

nþkþ2

Þð1  bcq Þð1  bq Þð1  abcq

2kþ1 2kþ2 bcq ; bcq ;q 2 ! kþ3þn kþ1þn ½n  kq ð1  bcq Þ ½n  k  1q ð1  bcq Þ 1 ;    2nþ3 2nþ1 aq 1  abcq 1  bcaq

rþn;kþ1 ¼ cq2knþ2 ½n  kq

ð1  bcq

Þ

! Þ

1  abcq

½kq ½n  k þ 1q ð1  bq Þ nþkþ1

 acqnkþ1

;

Þ

½n  k þ 1q ð1  bcq k

 ½n  kq þ

2nþ3

Þð1  abcq

Þ

! Þ

;

;

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802 nþkþ1

rþnþ1;k1 ¼ a2 bc q2nkþ4 ½kq

ð1  cqk Þð1  bcq

2nþ2 abcq ;q

Þ

;

2

nþkþ2

rþnþ1;k

nkþ2

¼ ac q

nþkþ2

ð1  bcq Þð1  abcq

2nþ2 abcq ;q

Þ

k

½n  kq þ

½kq ½n  k þ 1q ð1  bq Þ 1  bcq

2k



½k þ 1q ½n  kq ð1  bq

kþ1

2kþ2

1  bcq

! Þ

;

2

kþ1

rþnþ1;kþ1

¼ ½n  kq cq

kþ1

ð1  bq





kþ1 nþkþ2 nþkþ2 Þð1  bcq Þ bcq ;q abcq ;q 2 2

: 2kþ1 2kþ2 2nþ2 bcq ; bcq ; abcq ;q 2

Proof. Formulas (4.8) and (4.9) as well as (4.10) and (4.11), mentioned below, are obtained in a way, which we explain in case of the structure relation (4.8). Using the power representation (4.4) of p n;k ðx; y; a; b; cjqÞ and the q-differentiation rule Dq;x xi ¼ ½iq xi1 , we compute the coefficients ci;j in

xðx  yÞDq;x p n;k ðx; y; a; b; cj qÞ ¼

kþ1 nkþ1 X X i¼0

ci;j xi ykþji :

j¼0

Then we use the inverse representation formula (4.6) to obtain

xðx  yÞDq;x p n;k ðx; y; a; b; cj qÞ ¼

kþ1 nkþ1 X X i¼0

¼

kþj minði;lÞ X X

al;m ði; k þ j  iÞp l;m ðx; y; a; b; c jqÞ

l¼0 m¼0

j¼0

kþ1 nkþ1 X X i¼0

ci;j ci;j

nþ1 X nþ1 X al;m ði; k þ j  iÞp l;m ðx; y; a; b; cj qÞ: l¼0 m¼0

j¼0

Changing the order of summation gives

xðx 

yÞDq;x p n;k ðx; y; a; b; cj qÞ

¼

nþ1 X nþ1 X kþ1 nkþ1 X X l¼0 m¼0

i¼0

! ci;j al;m ði; k þ j  iÞ p l;m ðx; y; a; b; cj qÞ:

j¼0

By Theorem 3.3 (in particular, see (3.25)), we can reduce the range of summation of the first two sums, so that

xðx  yÞDq;x p n;k ðx; y; a; b; cj qÞ ¼

nþ1 kþ1 X X

sþl;m p l;m ðx; y; a; b; cj qÞ;

l¼n1 m¼k1

where

sþl;m :¼

kþ1 nkþ1 X X i¼0

ci;j al;m ði; k þ j  iÞ ¼

j¼0

kþ1 nkþ1 X X

ci;j al;m ði; k þ j  iÞ:

i¼m j¼lk

In the last step, the range of summation is reduced thanks to the fact that the coefficients ak;l ði; jÞ vanish for l > i or k > i þ j (cf. (4.7)); the resulting expression is a sum of at most nine terms. Now, the coefficients sþ can be computed using a coml;m puter algebra system. Actually, we have used MapleTM system to produce formulas given in the theorem. h Remark 2. On the Web page http://www.ii.uni.wroc.pl/ pwo/programs.html, one can find a MapleTM worksheet to produce þ   explicit expressions for the coefficients sþ l;m and r l;m in Eqs. (4.8) and (4.9) as well as the coefficients sl;m and r l;m in two other structure relations,

xðx  cqyÞDq1 ;x p n;k ðx; y; a; b; cj qÞ ¼

nþ1 kþ1 X X

sl;m p l;m ðx; y; a; b; cj qÞ;

ð4:10Þ

l¼n1 m¼k1

ðx  yÞðaq  yÞDq1 ;y p n;k ðx; y; a; b; cj qÞ ¼

nþ1 kþ1 X X

r l;m p l;m ðx; y; a; b; cj qÞ:

l¼n1 m¼k1

ð4:11Þ

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S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

4.1. Limit form: bivariate Jacobi polynomials Notice that

lim p n;k ðx; y; qa ; qb ; qc j qÞ ¼ q!1

ða;b;cÞ

where P n;k

ða;b;cÞ

Pn;k

ð1Þk k!ðn  kÞ! ðaþ1=2;bþ1=2;cþ1=2Þ P ð1  y; xÞ; ðb þ 1Þk ð2k þ b þ c þ 2Þnk n;k

ðx; yÞ are the triangle Jacobi polynomials (see [7, p. 86], or [16]), ð2kþbþc;a12Þ

ðx; yÞ :¼ Rnk

ðc12;b12Þ

ðxÞ ð1  xÞk Rk

y ; 1x

where a; b; c >  12, and

Rðml;mÞ ðtÞ :¼

   m; m þ l þ m þ 1  ðl þ 1Þm 1  t 2F1  m! lþ1

is the mth shifted Jacobi polynomial in one variable [15, Section 9.8]. Corollary 4.2. The following structure relations hold:

9 nþ1 kþ1 X X > @ ða;b;cÞ Hl;m > ða;b;cÞ > xðx þ y  1Þ P n;k ðx; yÞ ¼ sl;m Pl;m ðx; yÞ; > > = @x H n;k l¼n1 m¼k1

ð4:12Þ

nþ1 kþ1 > X X > @ ða;b;cÞ Hl;m ða;b;cÞ > Pn;k ðx; yÞ ¼ rl;m Pl;m ðx; yÞ; > yðx þ y  1Þ > ; @y H n;k l¼n1 m¼k1

where

Hi;j :¼ 

ð1Þj j!ði  jÞ!  ; b þ 12 j ð2j þ d þ 1Þij

sn1;k1

   k k þ c  12 n  k þ a þ 12   ¼ ; 2n þ k  12 2

sn;k1 ¼

   k k þ c  12 2k þ d  a  32     ; 2n þ k  12 2n þ k þ 32

snþ1;k1 ¼ 

sn1;k

sn;k

  k k þ c  12 ðn þ k þ dÞ   ; 2n þ k þ 12 2

    ¼ kð2n þ a þ 2c þ k þ 1Þðk þ dÞ þ ðd  1Þ c þ 12 n þ k þ 12

  ðk  nÞ n  k þ a  12   ; ð2k þ d  1Þð2k þ d þ 1Þ 2n þ k  12 2

( ) ðn  kÞðn þ k þ dÞ ðn  k þ 1Þðn þ k þ d þ 1Þ ¼  þ1 2n þ k  12 2n þ k þ 32      kðn  k þ 1Þ k þ b  12 ðk þ 1Þðn  kÞ k þ b þ 12   nkþ 2k þ d  1 2k þ d þ 1       ðk þ 1Þ k þ b þ 12 ðn  kÞðn þ k þ dÞ k k þ b  12   þ 1 ; 2k þ d  1 2k þ d þ 1 2n þ k  12

snþ1;k ¼

      ðn þ k þ d þ 1Þ n þ k þ k þ 12 kðn  k þ 1Þ k þ b  12 ðk þ 1Þðn  kÞ k þ b þ 12    ; n  k þ 2k þ d  1 2k þ d þ 1 2n þ k þ 12 2

sn1;kþ1 ¼

sn;kþ1 ¼

     ðn  k  1Þ2 k þ b þ 12 ðk þ dÞ k þ n þ k þ 12 n  k þ a  32 2   ; ð2k þ dÞ2 ð2k þ d þ 1Þ2 2n þ k  12 2

     ðk  nÞðk þ dÞðn þ k þ d þ 1Þ k þ b þ 12 n þ k þ k þ 12 2k þ k þ 32 n  k þ a  12     ; ð2k þ dÞ2 ð2k þ d þ 1Þ2 2n þ k  12 2n þ k þ 32

S. Lewanowicz et al. / Applied Mathematics and Computation 219 (2013) 8790–8802

snþ1;kþ1 ¼

8801

    ðk  nÞ k þ b þ 12 ðk þ dÞðn þ k þ d þ 1Þ2 n þ k þ k þ 12 2   ð2k þ dÞ2 ð2k þ d þ 1Þ2 2n þ k þ 12 2

and

rn1;k1

  k k þ c  12 ðk þ dÞ   ¼ ; 2n þ k  12 2

rn;k1 ¼  

rnþ1;k1 ¼

rn1;k ¼

rn;k ¼

  2k k þ c  12 ðk þ dÞ   ; 2n þ k  12 2n þ k þ 32

  k k þ c  12 ðk þ dÞ   ; 2n þ k þ 12 2

  kðk þ dÞðc  bÞðn  kÞ n  k þ a  12   ; ð2k þ d  1Þð2k þ d þ 1Þ 2n þ k  12 2

! ðc  bÞkðk þ dÞ ðn  kÞðn þ k þ dÞ ðn  k þ 1Þðn þ k þ d þ 1Þ ;  ð2k þ d  1Þð2k þ d þ 1Þ 2n þ k  12 2n þ k þ 32

rnþ1;k ¼

  ðc  bÞkðk þ dÞðn þ k þ d þ 1Þ n þ k þ k þ 12   ; ð2k þ d  1Þð2k þ d þ 1Þ 2n þ k þ 12 2

rn1;kþ1 ¼ 

rn;kþ1 ¼

    ðn  k  1Þ2 k k þ b þ 12 ðk þ dÞ n  k þ a  32 2   ; ð2k þ dÞ2 ð2k þ d þ 1Þ2 2n þ k  12 2

      2kðn  kÞ k þ b þ 12 ðk þ dÞ n þ k þ k þ 12 ðn þ k þ d þ 1Þ n  k þ a  12     ; ð2k þ dÞ2 ð2k þ d þ 1Þ2 2n þ k  12 2n þ k þ 32

rnþ1;kþ1 ¼ 

  kðk þ b þ 12Þðk þ dÞðn þ k þ d þ 1Þ2 n þ k þ k þ 12 2 ð2k þ dÞ2 ð2k þ d þ 1Þ2 ð2n þ k þ 12 Þ2

;

with k :¼ a þ b þ c; d :¼ b þ c. Remark 3. Structure relations (4.12) for the triangle Jacobi polynomials are in agreement with general results on classical orthogonal polynomials in two variables [5,10]. To our best knowledge, however, they were not given in full detail before. Acknowledgments The authors thank the referee for the valuable critical remarks and for some references which had not been considered in the first version of the paper. References [1] W.A. Al-Salam, T.S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3 (1972) 65–70. [2] M. Álvarez de Morales, L. Fernández, T.E. Pérez, M.A. Piñar, A semiclassical perspective on multivariate orthogonal polynomials, J. Comput. Appl. Math. 214 (2008) 447–456. [3] M. Álvarez de Morales, L. Fernández, T.E. Pérez, M.A. Piñar, Semiclassical orthogonal polynomials in two variables, J. Comput. Appl. Math. 207 (2007) 323–330. [4] G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. [5] I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Bivariate second-order linear partial differential equations and orthogonal polynomial solutions, J. Math. Anal. Appl. 387 (2012) 1188–1208. [6] C.F. Dunkl, Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Algebra. Discr. Methods 1 (1980) 137–151. [7] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. [8] L. Fernández, F. Marcellán, T.E. Pérez, M.A. Piñar, Recent trends on two variable orthogonal polynomials, in: Differential Algebra, Complex Analysis and Orthogonal Polynomials, Contemporary Mathematics, vol. 509, The American Mathematical Society, Providence, RI, 2010, pp. 59–86. [9] L. Fernández, T.E. Pérez, M.A. Piñar, Classical orthogonal polynomials in two variables: a matrix approach, Numer. Algorithms 39 (2005) 131–142. [10] L. Fernández, T.E. Pérez, M.A. Piñar, Weak classical orthogonal polynomials in two variables, J. Comput. Appl. Math. 178 (2005) 191–203. [11] L. Fernández, T.E. Pérez, M.A. Piñar, On Koornwinder classical orthogonal polynomials in two variables, J. Comput. Appl. Math. 236 (2012) 3817–3826. [12] A.G. Garcı`a, F. Marcellán, L. Salto, A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995) 147–162.

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