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Some properties of generalized multiple Hermite polynomials
Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
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Some properties of generalized multiple Hermite polynomials Cem Kaanoğlu a , M. Ali Özarslan b,∗ a
Cyprus International University, Faculty of Engineering, Lefkoşa, Mersin 10, Turkey
b
Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey
article
abstract
info
MSC: 33C45 42C05
The purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials. © 2011 Elsevier B.V. All rights reserved.
Keywords: Rodrigues formula Operational formula Generating function Multiple Hermite polynomials
1. Introduction The classical Hermite polynomials Hn (x) of degree n are given by the Rodrigues formula [1] 2
2
Hn (x) = (−1)n ex Dn e−x ,
D=
d
.
dx It is known that these polynomials are orthogonal over the interval (−∞, ∞) with respect to the weight function 2
w(x) = e−x , so that ∫ ∞ 2 Hn (x)e−x xk dx = 0,
k = 0, 1, . . . , n − 1.
−∞
− →
− →
→ (δ,− α)
Now, for the multi-index n = (n1 , . . . , nr ) ∈ Nr and α = (α1 , . . . , αr ), the multiple Hermite polynomials H− →
− → degree n = n1 + · · · + nr were given in [2] by the orthogonality relation
∫
∞
→ (δ,− α)
H− → −∞
n
(x)xk wi (x)dx = 0,
k = 0, 1, . . . , ni − 1; i = 1, 2, . . . , r
with respect to r weight functions
wi (x) = e
δ x2 +α x i
2
,
1 ≤ i ≤ r , δ < 0 and αi ̸= αj for i ̸= j.
→ (δ,− α)
The Rodrigues formula for the multiple Hermite polynomials H− → n
→ (δ,− α) H− (x) → n
= wr
−1
(wr wr−−11 (· · · (w2 w1−1 (w1 )(n1 ) )(n2 )
δ 2
= e− 2 x e−αr x ∗
d
nr
dxnr
eαr x e−αr −1 x · · ·
n2
d
dxn2
···)
(x) is defined by
(nr −1 ) (nr )
eα2 x e−α1 x
)
d
n1
dxn1
δ 2 e 2 x eα 1 x .
Corresponding author. Tel.: +90 392 630 1266; fax: +90 392 365 1604. E-mail addresses: [email protected] (C. Kaanoğlu), [email protected] (M. Ali Özarslan).
0377-0427/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2011.02.002
n
(x) of
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
4879
There have been many studies on polynomials via the Rodrigues formula [3–7] and new papers constantly coming out related with these generalizations [8–12]. Very recently, the authors [13] generalized the multiple Laguerre I and II polynomials [14] via the Rodrigues formula and obtained their explicit formulas, some operational formulas and the corresponding generating functions. For the multiple Laguerre I polynomials, they considered the following generalization
(−1)n1 +n2 Kn(α1 ,1n,α2 2 ) (x; k; p) = epx x−α1
dn 1
k
dxn1
xα1 +n1 −α2
dn2 dxn2
xα2 +n2 e−px
k
(1)
and they obtained their explicit form as 2) (x; k; p) = (−1)n1 +n2 n1 !n2 ! Kn(α1 ,1n,α 2
n2 − i − pi i =0 j =0
n1
×
l
−− p
l
l! l =0 h =0
(−1)h
i!
(−1)j
i
α2 + n2 + kj
j
n2
l
α1 + n1 + ki + kh
h
n1
xki
xkl .
(2)
In this paper we introduce the following family of polynomials dn2 dn1 k k 1 ,α2 ) G(α,α (x; k; p) = x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x n1 ,n2 dx 1 dx 2
(3)
where p > 0; α, α1 , α2 > −1 with α1 ̸= α2 and k is a natural number. It should be noticed that, in the particular case (δ,α ,α ) k = 2, α = 0 and p = − 2δ , Relation (3) gives the Rodrigues formula of the multiple Hermite polynomials Hn1 ,n21 2 (x) given in [2] and in the particular case k = 1 and p = 1, definition (1) gives the Rodrigues formula of the multiple Laguerre I polynomials [14]. We organize the paper as follows. In Section 2, we obtain the explicit forms of the polynomials as
n1 n2 −r − n2 i − α1l xl − pi i α + kj ki−n2 α2r xr − (−1)j x r ! l=0 l! i=0 j=0 i! j n2 − r r =0 n l1 1 −l − − pl1 l1 α − n2 + r + ki + kh kl1 −n1 × (−1)h x l ! h n1 − l l =0 h=0 1
1 ,α2 ) G(α,α (x; k; p) = n1 !n2 ! n1 ,n2
1
(α,α ,α2 )
which shows that the polynomials Gn1 ,n21
(x; k; p) are of exact degree (k − 1)(n1 + n2 ). (α,α ,α )
(α ,α )
In Section 3, we obtain an operational formula for the polynomials Gn1 ,n21 2 (x; k; p) in terms of Kn1 ,1n2 2 (x; k; p). In Section 4, the linear generating functions and a recurrence formula are obtained. In Section 5, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships for the extended multiple Hermite polynomials (α,α ,α ) Gn1 ,n21 2 (x; k; p). In the last section, some applications are given. 2. Explicit expression of the polynomials Chatterjea [4] defined the generalized Laguerre polynomials via the Rodrigues formula by 1 −α pxk n α+n −pxk x e D (x e ); p, k ∈ N, n! and using the following generalized rule for the derivative of a sufficiently differentiable function of x [15], (α)
Tkn (x, p) =
D (f (x)) = s
s − (−1)k k=0
where Dn = (α)
dn , dxn
k!
k − k j D f ( x) (−1) xk−j Ds xj , k
j
j =0
(4)
(5)
he obtained the explicit formula of the polynomials as
Tkn (x, p) =
n − pi i=0
i!
i − i α + n + kj j x (−1) . ki
j
j=0
n
(6)
Furthermore, in [4], Chatterjea derived the formula Dn (xα+n e−px Y ) = n!xα e−px k
k
n − xr
r! r =0
(α+r )
Tk(n−r ) (x, p)Dr Y
where Y is any sufficiently differentiable function of x.
(7)
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C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
Now, using (3), (6) and (7), we get the following k k 1 ,α2 ) (x; k; p) = x−α epx e−α1 x Dn1 eα1 x e−α2 x Dn2 xα e−px eα2 x G(α,α n 1 ,n 2
= n2 !x−α epx e−α1 x Dn1 eα1 x e−α2 x xα−n2 e−px k
k
n2 − xr
r! r =0
(α−n +r )
Tk(n −2r ) (x, p)Dr (eα2 x ) 2
n2 n2 −r i − i − 1 − p i α + kj k (−1)j α2r Dn1 (xα+r +ki−n2 e−px eα1 x ) r ! i ! j n − r 2 r =0 i =0 j =0 n− n2 i 2 −r i − − 1 p i α + kj j = n1 ! n2 ! (−1) α2r r ! i =0 i ! j =0 j n2 − r r =0 k
= n2 !x−α epx e−α1 x
× x−n2 −n1 +r +ki
n1 − xl l=0
l!
(α−n +r +ki−n1 +l)
Tk(n −2l) 1
(x, p)α1l
− n2 n2 −r i − n1 i − 1 − 1 p i α + kj j = n1 ! n2 ! (−1) α2r r ! i ! j n − r l ! 2 r =0 l =0 i =0 j =0 n l1 1 −l l 1 − − p l1 α − n2 + r + ki + kh × (−1)h α1l x−n2 −n1 +r +l+ki+kl1 . l ! h n − l 1 1 l =0 h=0 1
Then, we can state the following theorem, which gives the explicit form of the polynomials defined by (3). (α,α ,α2 )
(x; k; p) has the explicit expression i 2 −r − − α r xr − α l xl n− pi i α + kj ki−n2 1 2 j 1 ,α2 ) G(α,α (− 1 ) x ( x ; k ; p ) = n ! n ! 1 2 n 1 ,n 2 r ! l=0 l! i=0 j=0 i! j n2 − r r =0
Theorem 2.1. The family of polynomials Gn1 ,n21 n1
n2
n 1 −l
×
l1 −− pl1
l1 =0 h=0
l1 !
(−1)h
l1 h
α − n2 + r + +ki + kh kl1 −n1 x . n1 − l
(8)
Comparing (2) and (8), it can be stated that: (α,α ,α2 )
Corollary 2.2. The family of polynomials Gn1 ,n21 1 ,α2 ) G(α,α (x; k; p) = n 1 ,n 2
n2
−
(x; k; p) has the explicit expression
n1
α2r xr
−
r =0
α1l xl (−1)n1 −l+n2 −r
l=0
(0,α ,α2 )
n n 1 2 l
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
r
(x; k; p)x−n1 −n2 .
(δ,α ,α )
x; 2; − 2δ = Hn1 ,n21 2 (x), then as a consequence of Corollary 2.2 we (α ,α ) have the following relation between the multiple Hermite polynomials and the polynomials Kn1 ,1n2 2 (x; k; p): Remark 2.1. It should be noticed that, since Gn1 ,n12
1 ,α2 ) Hn(δ,α (x) = 1 ,n2
n2 −
α2r xr
r =0
n1 −
α1l xl (−1)n1 −l+n2 −r
n n 1 2
l =0
l
r
(−n +r −n +l,−n2 +r )
K(n −2l,n −r1) 1 2
x; 2; −
δ 2
x−n1 −n2 .
3. Operational formulas In [4], Chatterjea obtained the operational formula x−α epx Dn (xα+n e−px Y ) = k
k
n ∏
(xD − pkxk + α + j)Y
(9)
j =1
where Y is any sufficiently differentiable function of x. Now use Leibniz’ formula to find dn 1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y dx 1 dx 2 k
= x−α epx e−α1 x
dn1 dxn1
eα1 x e−α2 x
n2 − n2 s=0
s
Dn2 −s (eα2 x )Ds (xα e−px Y ) k
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
4881
then with the help of Eq. (9), we get dn 2 dn1 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y 1 2 dx dx dn 1
k
= x−α epx e−α1 x k
= x−α epx e−α1 x
dxn1
eα1 x e−α2 x
=x
e
e
n2 − n2
k
n2 − n2
s
×
dn 1
α
n −s
s ∏
(xD − pkxk + α − s + j)Y
j =1
n1 − n1
l
n1 − n1
l
l =0
k
n −s
k
s ∏
(xD − pkxk + α − s + j)Y
j =1
l=0
α2 2
α2 2 eα2 x xα−s e−px
eα1 x xα−s e−px
dxn1
n 2 −s 2
s
n2 − n2 s=0
l ∏
n −s
α2 2
s
s=0
= x−α epx
s
s=0
s=0
−α pxk −α1 x
n2 − n2
(xD − pkxk + α − s − l + i)
D
n1 −l
(e
α1 x
)D
l
x
α−s −pxk
e
s ∏
(xD − pkx + α − s + j)Y k
j=1
α1 1 xα−s−l e−px n −l
k
s ∏ (xD − pkxk + α − s + j)Y .
i =1
j =1
Hence, we obtain the first equality k
x−α epx e−α1 x
=
dn1 dxn1
n2 − n2 s=0
s
eα1 x e−α2 x
n −s
α2 2
dn 2
n1 − n1
l
l=0
xα e−px eα2 x Y k
dxn2
n −l
α1 1 x−s−l
l ∏
(xD − pkxk + α − s − l + i)
i =1
s ∏
(xD − pkxk + α − s + j)Y .
j =1
Now, taking (7) into consideration, we have dn1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y 1 2 dx dx k
= n2 !x−α epx e−α1 x k
= n2 !x−α epx e−α1 x n 2 −r
×
dn1 dxn1 dn1 dxn1
eα1 x e−α2 x xα−n2 e−px
k
eα1 x e−α2 x xα−n2 e−px
k
i − pi − xki (−1)j i! i=0 j =0
i
α + kj
j
n2 − r
n2 − xr
r! r =0
(α−n +r )
Tk(n −2r ) (x, p)Dr (eα2 x Y ) 2
n2 − xr
r! r =0 − r r
r 1 =0
r1
r −r1 α2 x
α2
e
Dr1 (Y )
r n2 n2 −r i − i − 1 − p i α + kj − r k r −r (−1)j α2 1 Dn1 (xα−n2 +r +ki e−px eα1 x Dr1 (Y )) r ! i =0 i ! j =0 j n2 − r r = 0 r 1 r =0 1 n n − r i 2 2 i − − − p i α + kj 1 k = n1 !n2 !x−α epx e−α1 x (−1)j r ! i=0 i! j=0 j n2 − r r =0 n r 1 − − r xl (α−n2 +r +ki−n1 +l) k r −r × α2 1 xα−n2 +r +ki−n1 e−px Tk(n −l) (x, p)Dl (eα1 x Dr1 (Y )) 1 l ! r 1 r 1 =0 l =0 r n− n2 i 2 −r i − − 1 p i α + kj − r k r −r j −α pxk −α1 x = n1 !n2 !x e e (−1) α2 1 xα−n2 +r +ki−n1 e−px r ! i ! j n − r r 2 1 i=0 j =0 r 1 =0 r =0 n n − l l l 1 1 1 − xl − pl1 kl1 − l1 α − n2 + r + ki + kh − l × x (−1)h Dl−h1 (eα1 x )Dh1 (Dr1 (Y )) l ! l ! h n − l h 1 1 1 l =0 l 1 =0 h=0 h1 =0 n− n2 n1 l1 i l l n− r r − 2 −r 1 −l l1 i − − α2 x α1 x p ki i α + kj p kl1 − l1 j h = n2 !n1 ! x (−1) x (−1) r ! l ! i ! j n − r l ! h 2 h=0 r =0 l=0 i=0 j =0 l1 =0 1 r l α − n2 + r + ki + kh −n1 −n2 − r 1 − l 1 h 1 +r 1 × x D (Y ). r1 n1 − l r1 α2 h =0 h1 α h1 r1 =0 1 1 k
= n2 !x−α epx e−α1 x
(10)
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Hence, the second equality will be dn 1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y dx 1 dx 2
=
n2 −
α2r xr
n1 −
α1l xl (−1)n1 −l+n2 −r
n n 1 2 l
l =0
r =0
r
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
(x, k, p)
l r − l 1 h 1 +r 1 r 1 − (Y )x−n1 −n2 . D × r1 r1 α2 h =0 h1 α h1 r1 =0 1 1
(11)
Thus, it follows from (10) and (11). (α ,α )
Theorem 3.1. Let Kn1 ,1n2 2 (x; k; p) be defined by (2). Then we have n2 − n2
r
r =0
=
n2 − r =0
×
n −r
α2 2
n1 − n1
l
l =0
α2r xr
n1 −
n −l
α1 1 x−r −l
l ∏
(xD − pkxk + α − r − l + i)
i=1
α1l xl (−1)n1 −l+n2 −r
(xD − pkxk + α − r + j)Y
j =1
n n 1 2 l
l =0
r ∏
r
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
(x; k; p)
r l − r 1 − l 1 h 1 +r 1 D (Y )x−n1 −n2 . r1 h1 r h α 1 1 α 2 h 1 =0 r1 =0 1
(12)
Setting Y = 1 in (12), we obtain the following theorem. (α,α ,α2 )
Theorem 3.2. For Gn1 ,n21
1 ,α2 ) G(α,α (x; k; p) = n 1 ,n 2
(x; k; p), we have
n2 − n2
r
r =0
n −r
α2 2
n1 − n1
l
l =0
n −l
α1 1 x−r −l
l
×
∏
(xD − pkxk + α − r − l + i)
r ∏
i=1
(xD − pkxk + α − r + j).
(13)
j=1
Furthermore, letting k = 2, p = − 2δ and α = 0 in (13), we get the following result. (δ,α ,α2 )
Corollary 3.3. For the multiple Hermite polynomials Hn1 ,n21 1 ,α2 ) Hn(δ,α (x) = 1 ,n2
n2 − n2 r =0
r
n −r
α2 2
n1 − n1 l =0
l
n −l
α1 1 x−r −l
(x), we have
l ∏
(xD + δ x2 − r − l + i)
i=1
r ∏
(xD + δ x2 − r + j).
j=1
Moreover, for each suitable choice of the sufficiently differentiable function Y , Theorem 3.1 can be shown to yield various interesting operational formulas for the polynomials defined by (3) and of course for the multiple Hermite polynomials. 4. Generating functions and recurrence relations In this section we obtain the generating functions and a recurrence relation for a class of polynomials defined by (3). The first result of this section is the following theorem: (α,α ,α2 )
Theorem 4.1. Let Gn1 ,n21 G∗ (x; t1 ; t2 ) =
(x; k; p) be the polynomials defined by (3). Then we have
∞ − n 1 ,n 2 = 0
n
1 ,α2 ) Gn(α,α (x; k; p) 1 ,n 2
n
t2 2 t1 1 n2 ! n1 !
= x−α (x + t1 + t2 )α eα1 t1 +α2 t2 +px Proof. Direct computations yield that
k −p(x+t
k 1 +t2 )
.
(14)
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
∞ − n1 ,n2 =0
=x
1 ,α2 ) G(α,α (x; k; p) n1 ,n2
−α pxk −α1 x
e
e
=x
e
e
k
= x−α epx e−α1 x k
k
= x−α epx e−α1 x k
= x−α epx e−α1 x
dxn1
∞ − dn1 n1 =0
= x−α epx e−α1 x
dxn1
∞ − dn1 n1 =0
4883
n t1 1
n2 ! n1 !
∞ − dn1 n1 =0
−α pxk −α1 x
n t2 2
dxn1
e
α1 x −α2 x
e
∞ − dn 2 n 2 =0
e
α1 x −α2 x
e
1
dxn2
eα1 x e−α2 x
1
x e
n2 !
z−x
C1
zα e
2π i
e
C1
−pz k
eα 2 z
z − x − t2
n
t1 1 n1 !
k ∞ z α e−pz eα2 z −
2π i
n2
α −pxk α2 x t2
dz
dz
z−x
n 2 =0
n2
t2
n
t1 1 n1 !
n
t1 1 n1 !
n1
∞ − dn1
t k eα1 x (x + t2 )α e−p(x+t2 ) eα2 t2 1 n1 dx n 1! n1 =0 k ∞ eα1 η (η + t2 )α e−p(η+t2 ) eα2 t2 − t1
1 2π i
2π i
n 1 =0
k eα1 η (η + t2 )α e−p(η+t2 ) eα2 t2
1
η−x
C2
η − x − t1
C2
n1
η−x
dη
dη
where C1 is a circle in the complex z-plane, cut along the negative real axis, (centered at z = x + t2 ) with radius ε > 0, which is described in the positive direction (counter-clockwise) and C2 is a circle (centered at η = x + t1 ) in the complex
t1 η-plane, cut along the negative real axis, with sufficiently small radius and zt−2 x < 1, η− < 1. Thus x n
∞ − n1 ,n2 =0
1 ,α2 ) G(α,α (x; k; p) n1 ,n2
n
t2 2 t1 1 n2 ! n1 !
= x−α (x + t1 + t2 )α eα1 t1 +α2 t2 +px
k −p(x+t
1 + t2 )
k
.
Remark 4.1. Letting k = 2, p = − 2δ and α = 0 in (14), we get the generating function for the multiple Hermite polynomials which was already obtained in [2, Eq (6)]. (α,α ,α2 )
Theorem 4.2. For Gn1 ,n21
(x; k; p), we have
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) = 1 ,n2
n1 − n2 − n1 n2 l 1 =0 l 2 =0
l1
l2
(α,α ,α )
Gn1 −l11 ,n22 −l2 x; k;
p (β,β1 ,β2 ) p Gl1 ,l2 . x; k; 2 2
Proof. Taking (14) into consideration, we have ∞ − ∞ −
n
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) 1 ,n 2
n 1 =0 n 2 =0
n
t2 2 t1 1 n2 ! n1 !
= x−(α+β) (x + t1 + t2 )(α+β) e(α1 +β1 )t1 +(α2 +β2 )t2 +px −p(x+t1 +t2 ) n +l n +l ∞ ∞ − − p (β,β1 ,β2 ) p t1 1 1 t2 2 2 1 ,α2 ) = G x ; k ; . Gn(α,α x ; k ; , n l , l 1 2 1 2 2 2 n1 !l1 !n2 !l2 ! n ,n = 0 l ,l = 0 k
1
2
k
1 2
Taking n1 −→ n1 − l1 and n2 −→ n2 − l2 , we get ∞ − ∞ −
n
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) 1 ,n 2
n 1 =0 n 2 =0
=
n1 − n2 ∞ − − n1 ,n2 =0 l1 =0 l2 =0
Hence the result.
(α,α ,α )
n
t2 2 t1 1 n2 ! n1 !
Gn1 −l11 ,n22 −l2 x; k;
n
n
p (β,β1 ,β2 ) p t1 1 t2 2 Gl1 ,l2 x; k; . 2 2 (n1 − l1 )!l1 !(n2 − l2 )!l2 !
Letting k = 2, p = − 2δ and α = 0 in (15), we get the following recurrence relation.
(15)
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C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
Corollary 4.3. The recurrence relation for the multiple Hermite polynomials is 1 +β1 ,α2 +β2 ) Hn(δ,α (x) ,m
=
n − m − n m
k
k=0 l=0
l
δ ,α ,α 1 2
δ ,β ,β 1 2
Hn−k,m−l (x)Hk,l 2
2
(x).
5. Multilateral generating functions Very recently several substantially more general families of bilinear, bilateral, and mixed multilateral finite-series relationships and generating functions for the multiple Laguerre and multiple Hermite polynomials are given in [16]. In this section, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships for the extended (α,α ,α ) multiple Hermite polynomials Gn1 ,n21 2 (x; k; p) by applying the same method used in [17,16]. We start with the following theorem. Theorem 5.1. Corresponding to an identically nonvanishing function Φµ(1) (ξ1 , ξ2 , . . . , ξs ) of s (real or complex) variables ξ1 , ξ2 , . . . , ξs (sϵ N = N0 \ {0}) and of (complex) order µ, let
∆(1) (ξ1 , ξ2 , . . . , ξs ; w) :=
∞ −
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )w l
al ̸= 0.
(16)
l =0
Suppose also that
n
k
(1) N2 N1 − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )
Θn,N1 ,N2 (x; ξ1 , ξ2 , . . . , ξs ; ζ , ρ) :=
k=0
l =0
(n − N2 k)!(k − N1 l)!
(α+λl,α +λ l,α2 +λ2 l)
× Gn−N2 k,k1−N11l
(x; k; p)ζ k ρ l (nϵ N)
(17)
then ∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
=∆
(1)
ξ1 , ξ2 , . . . , ξs ;
t2
,
N t1 2
γ
t1n
N t2 1
γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 )
xλ
G∗ (x; t1 ; t2 )
(18)
provided that each member of (18) exists. The notation
n q
means that the greatest integer is less than or equal to nq .
Proof. Considering (17) into (18), we get
∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
t2 N
,
γ
t1n =
N
t1 2 t2 1
n
k
(1) N2 N1 ∞ − − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) n=0 k=0
l=0
(n − N2 k)!(k − N1 l)! k
(α+λl,α +λ l,α +λ l) × Gn−N2 k,k1−N11l 2 2 (x; k; p)
t2
γ
N
t2 1
t1 2
N
l t1n .
Using n −→ n + N2 k
∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
t2 N
,
γ
=
N
t1 2 t2 1
k
(1) N1 ∞ − ∞ − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) n=0 k=0 l=0
n!(k − N1 l)!
(α+λl,α +λ l,α +λ l) k−N l × Gn,k−N1 l1 1 2 2 (x; k; p)γ l t1n t2 1 .
Taking k −→ k + N1 l and using (14), we find ∞ − n =0
Θn,N1 ,N2
x; ξ1 , ξ2 , . . . , ξs ;
t2 N
,
γ N
t1 2 t2 1
=
∞ −
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )γ l x−(α+λl) (x + t1 + t2 )α+λl
l =0
× exp((α1 + λ1 l)t1 + (α2 + λ2 l)t2 + pxk − p(x + t1 + t2 )k )
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
∞ −
=
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )
γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 )
4885
l
xλ
l=0
× x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) G∗ (x; t1 ; t2 ). = ∆(1) ξ1 , ξ2 , . . . , ξs ; λ x
Theorem 5.2. Corresponding to an identically nonvanishing function Φµ(21),µ2 (ξ1 , ξ2 , . . . , ξs ) of s (real or complex) variables ξ1 , ξ2 , . . . , ξs (sϵ N = N0 \ {0}) and of (complex) order µ, let (2)
∆Ψ1 ,Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; τ1 , τ2 ) :=
∞ − ∞ −
(2)
k
k
ak1 ,k2 ̸= 0.
ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )τ1 1 τ2 2
k1 =0 k2 =0
Suppose also that
n1
n2
(2) q1 q2 − − ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
1 ,λ2 ,Ψ1 ,Ψ2 Λnµ,λ (x; ξ1 , ξ2 , . . . , ξs ; ζ1 , ζ2 ) := 1 ,n 2 ,q 1 ,q 2
(n1 − q1 k1 )!(n2 − q2 k2 )!
k1 =0 k2 =0
(α,α +λ k ,α +λ k2 )
× Gn1 −q11 k1 ,1n21−q22 k2 2
k
k
(x; k; p)ζ1 1 ζ2 2
(19)
then ∞ − n1 ,n2 =0
v1 v2 n n 1 ,λ2 ,Ψ1 ,Ψ2 , t1 1 t2 2 Λµ,λ x ; ξ , ξ , . . . , ξ ; 1 2 s q1 q2 n1 ,n2 ,q1 ,q2 t1
t2
= ∆(Ψ21),Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; v1 eλ1 t1 , v2 eλ2 t2 )G∗ (x; t1 ; t2 ).
(20)
Proof. Considering (19) into (20), we get
∞ −
n1 ,n2 =0
1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x; ξ1 , ξ2 , . . . , ξs ; n1 ,n2 ,q1 ,q2
v1 q
,
v2
n
n
t1 1 t2 2 =
q
t1 1 t2 2
n1
n2
(2) q1 q2 ∞ − − − ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
(n1 − q1 k1 )!(n2 − q2 k2 )! k1 k2 v1 v2 (α,α +λ k ,α +λ k ) n n × Gn1 −q11 k1 ,1n21−q22 k2 2 2 (x; k; p) q1 t1 1 t2 2 . q2 n1 ,n2 =0 k1 =0 k2 =0
t1
t2
Taking n2 −→ n2 + q2 k2 ∞ −
n1 ,n2 =0
1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x; ξ1 , ξ2 , . . . , ξs ; n1 ,n2 ,q1 ,q2
v1 q t1 1
,
v2
n
n
t1 1 t2 2 =
q t2 2
∞ −
n1 q1
(2) ∞ a −− k1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
n1 ,n2 =0 k1 =0 k2 =0
(n1 − q1 k1 )!n2 !
(α,α +λ k ,α +λ k ) k k n −q k n × Gn1 −q11 k1 ,1n21 2 2 2 (x; k; p)v11 v22 t1 1 1 1 t2 2 .
Taking n1 −→ n1 + q1 k1 and considering (14), we have ∞ − n1 ,n2 =0
=
v1 v2 n n 1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x ; ξ , ξ , . . . , ξ ; , t1 1 t2 2 1 2 s q1 q2 n1 ,n2 ,q1 ,q2
∞ − ∞ − k1 =0 k2 =0
=
∞ − ∞ −
t1
∞ − n1 ,n2 =0
t2
n1
t 1 +λ1 k1 ,α2 +λ2 k2 ) Gn(α,α (x; k; p) 1 1 ,n 2
n
t2 2
n1 ! n2 !
(2)
x−α (x + t1 + t2 )α exp((α1 + λ1 k1 )t1 + (α2 + λ2 k2 )t2 + pxk − p(x + t1 + t2 )k )
k1 =0 k2 =0
× ak1 ,k2 Φµ(21)+Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )v11 v22 ∞ − ∞ − (2) = ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )(v1 eλ1 t1 )k1 (v2 eλ2 t2 )k2 k
k
k
ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )v11 v22
k
k1 =0 k2 =0
× x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) = ∆(Ψ21),Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; v1 eλ1 t1 , v2 eλ2 t2 )G∗ (x; t1 ; t2 ).
4886
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
6. Applications of main results (1)
(2)
If the multivariable functions Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) and Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs ) can be selected by means of several functions of one variable then further applications of Theorems 5.1 and 5.2 can be given. It should be noted here that, one should use Theorem 5.1 in order to give bilinear, bilateral, mixed multilateral finite-series relationships and generating functions between the polynomials of which contain one summation symbol in its generating relation and the extended multiple Hermite polynomials. On the other hand, one cannot give any bilinear, bilateral, mixed multilateral finite-series relationships and generating functions between the polynomials of which contain more than one summation symbol in its generating relation and the extended multiple Hermite polynomials by using Theorem 5.1. But one can use Theorem 5.2 to achieve this problem. In this section we give the following illustrative examples. (1) Example 6.1. Letting s = 1, µ = 0, Ψ = 1, ξ1 = x, al = 1 and Φl (x) = Lαn (x), where Lαn (x) is the classical Laguerre polynomial (see [12]) and it is known that ∞ −
Lαn (x)t n = (1 − t )−α−1 exp(x(1 − (1 − t )−1 ))
n =0
then by Theorem 5.1, we obtain the generating function between the classical Laguerre polynomials and the extended multiple Hermite polynomials
n
k
(α+λl,α +λ l,α +λ l) N2 N1 ∞ − − − Lαn (x)Gn−N2 k,k1−N11l 2 2 (x; k; p) n=0 k=0
(n − N2 k)!(k − N1 l)!
l =0
t2 N
t1 2
k
γ N
t2 1
l t1n
= x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) −α−1 −1 γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) . × 1− exp x 1 − 1 − λ λ x
x
Choosing s = 1, ξ1 = x, µ1 = µ2 = 0, Ψ1 = Ψ2 = 1, ak1 ,k2 = Laguerre I polynomial (see [15]), in Theorem 5.2, we get k
1 k1 !k2 !
∞ − ∞ −
(α ,α ;β)
Lk1 1,k2 2
(α ,α ;β)
(x), where Lk1 1,k2 2
(x) is a
k
τ1 1 τ2 2 k1 ! k2 ! k1 =0 k2 =0 1 β(t1 + t2 − t1 t2 ) exp = x (1 − t1 )α1 (1 − t2 )α2 (1 − t1 )(1 − t2 )
(2)
∆1,1,0,0 (x; τ1 , τ2 ) :=
(α ,α ;β)
(2)
and Φk1 ,k2 (x) = Lk1 1,k2 2
(x)
for |t1 | < 1, |t2 | < 1
and therefore, the generating relation between Laguerre I polynomials and extended multiple Hermite polynomials is ∞ −
n1 q1
n2 q2
1 +λ1 k1 ,α2 +λ2 k2 ) − − Lk(α1,k,α2 ;β) (x)Gn(α,α (x; k; p) v1 k1 v2 k2 1 2 1 −q1 k1 ,n2 −q2 k2
n1 ,n2 =0 k1 =0 k2 =0
(n1 − q1 k1 )!(n2 − q2 k2 )!k1 !k2 !
q
t1 1
q
t2 2
n
n
t1 1 t2 2 1
= x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) (1 − v1 eλ1 t1 )α1 (1 − v2 eλ2 t2 )α2 λ1 t1 λ2 t2 λ1 t1 +λ2 t2 β(v1 e + v2 e − v1 v2 e ) x for v1 eλ1 t1 < 1, v2 eλ2 t2 < 1. × exp λ t λ t 1 1 2 2 (1 − v1 e )(1 − v2 e ) Acknowledgements The authors are very grateful to the anonymous referees for the careful reading of the paper. References [1] E.D. Rainville, Special Functions, The Macmillian Company, New York, 1960. [2] D.W. Lee, Properties of multiple Hermite and multiple Laguerre polynomials by the generating function, Integral Transforms Spec. Funct. 18 (2007) 855–869. [3] S.K. Chatterjea, A generalization of Laguerre polynomials, Collect. Math. 15 (1963) 285–292. [4] S.K. Chatterjea, On a generalization of Laguerre polynomials, Rend. Semin. Mat. Univ. Padova 34 (1964) 180–190. [5] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalization of Hermite polynomials, Duke Math. J. 29 (1962) 51–64.
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J.P. Singhal, C.M. Joshi, On the unification of generalized Hermite and Laguerre polynomials, Indian J. Pure Appl. Math. 13 (8) (1982) 904–906. H.M. Srivastava, J.P. Singhal, A class of polynomials defined by generalized Rodrigues’ formula, Ann. Mat. Pura Appl. 90 (1971) 75–85. L. Carlitz, A note on certain biorthogonal polynomials, Pacific J. Math. 24 (1968) 425–430. K.Y. Chen, A new summation identity for the Srivastava–Singhal polynomials, J. Math. Anal. Appl. 298 (2004) 411–417. K.Y. Chen, C.J. Chyan, H.M. Srivastava, Some polynomial systems associated with a certain family of differential operators, J. Math. Anal. Appl. 268 (2002) 344–377. K.Y. Chen, H.M. Srivastava, A limit relationship between Laguerre and Hermite polynomials, Integral Transforms Spec. Funct. 16 (2005) 75–80. H.M. Srivastava, Some families of generating functions associated with the Stirling numbers of the second kind, J. Math. Anal. Appl. 251 (2000) 752–769. M.A. Özarslan, C. Kaanoğlu, Some generalizations of multiple Laguerre polynomials via Rodrigues formula, Ars Combinatoria (in press). W.V. Assche, Els Coussement, Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001) 317–347. I.J. Schwatt, An Introduction to the Operations with Series, Philadelphia, 1924. M.A. Özarslan, E. Özergin, C. Kaanoğlu, Multilateral generating functions for the multiple Laguerre and multiple Hermite polynomials, J. Comput. Anal. Appl. 12 (2010) 667–677. H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press, Ellis Horwood Limited, Chichester, John Wiley and Sons, New York, 1984.
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Some properties of generalized multiple Hermite polynomials Cem Kaanoğlu a , M. Ali Özarslan b,∗ a
Cyprus International University, Faculty of Engineering, Lefkoşa, Mersin 10, Turkey
b
Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey
article
abstract
info
MSC: 33C45 42C05
The purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials. © 2011 Elsevier B.V. All rights reserved.
Keywords: Rodrigues formula Operational formula Generating function Multiple Hermite polynomials
1. Introduction The classical Hermite polynomials Hn (x) of degree n are given by the Rodrigues formula [1] 2
2
Hn (x) = (−1)n ex Dn e−x ,
D=
d
.
dx It is known that these polynomials are orthogonal over the interval (−∞, ∞) with respect to the weight function 2
w(x) = e−x , so that ∫ ∞ 2 Hn (x)e−x xk dx = 0,
k = 0, 1, . . . , n − 1.
−∞
− →
− →
→ (δ,− α)
Now, for the multi-index n = (n1 , . . . , nr ) ∈ Nr and α = (α1 , . . . , αr ), the multiple Hermite polynomials H− →
− → degree n = n1 + · · · + nr were given in [2] by the orthogonality relation
∫
∞
→ (δ,− α)
H− → −∞
n
(x)xk wi (x)dx = 0,
k = 0, 1, . . . , ni − 1; i = 1, 2, . . . , r
with respect to r weight functions
wi (x) = e
δ x2 +α x i
2
,
1 ≤ i ≤ r , δ < 0 and αi ̸= αj for i ̸= j.
→ (δ,− α)
The Rodrigues formula for the multiple Hermite polynomials H− → n
→ (δ,− α) H− (x) → n
= wr
−1
(wr wr−−11 (· · · (w2 w1−1 (w1 )(n1 ) )(n2 )
δ 2
= e− 2 x e−αr x ∗
d
nr
dxnr
eαr x e−αr −1 x · · ·
n2
d
dxn2
···)
(x) is defined by
(nr −1 ) (nr )
eα2 x e−α1 x
)
d
n1
dxn1
δ 2 e 2 x eα 1 x .
Corresponding author. Tel.: +90 392 630 1266; fax: +90 392 365 1604. E-mail addresses: [email protected] (C. Kaanoğlu), [email protected] (M. Ali Özarslan).
0377-0427/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2011.02.002
n
(x) of
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
4879
There have been many studies on polynomials via the Rodrigues formula [3–7] and new papers constantly coming out related with these generalizations [8–12]. Very recently, the authors [13] generalized the multiple Laguerre I and II polynomials [14] via the Rodrigues formula and obtained their explicit formulas, some operational formulas and the corresponding generating functions. For the multiple Laguerre I polynomials, they considered the following generalization
(−1)n1 +n2 Kn(α1 ,1n,α2 2 ) (x; k; p) = epx x−α1
dn 1
k
dxn1
xα1 +n1 −α2
dn2 dxn2
xα2 +n2 e−px
k
(1)
and they obtained their explicit form as 2) (x; k; p) = (−1)n1 +n2 n1 !n2 ! Kn(α1 ,1n,α 2
n2 − i − pi i =0 j =0
n1
×
l
−− p
l
l! l =0 h =0
(−1)h
i!
(−1)j
i
α2 + n2 + kj
j
n2
l
α1 + n1 + ki + kh
h
n1
xki
xkl .
(2)
In this paper we introduce the following family of polynomials dn2 dn1 k k 1 ,α2 ) G(α,α (x; k; p) = x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x n1 ,n2 dx 1 dx 2
(3)
where p > 0; α, α1 , α2 > −1 with α1 ̸= α2 and k is a natural number. It should be noticed that, in the particular case (δ,α ,α ) k = 2, α = 0 and p = − 2δ , Relation (3) gives the Rodrigues formula of the multiple Hermite polynomials Hn1 ,n21 2 (x) given in [2] and in the particular case k = 1 and p = 1, definition (1) gives the Rodrigues formula of the multiple Laguerre I polynomials [14]. We organize the paper as follows. In Section 2, we obtain the explicit forms of the polynomials as
n1 n2 −r − n2 i − α1l xl − pi i α + kj ki−n2 α2r xr − (−1)j x r ! l=0 l! i=0 j=0 i! j n2 − r r =0 n l1 1 −l − − pl1 l1 α − n2 + r + ki + kh kl1 −n1 × (−1)h x l ! h n1 − l l =0 h=0 1
1 ,α2 ) G(α,α (x; k; p) = n1 !n2 ! n1 ,n2
1
(α,α ,α2 )
which shows that the polynomials Gn1 ,n21
(x; k; p) are of exact degree (k − 1)(n1 + n2 ). (α,α ,α )
(α ,α )
In Section 3, we obtain an operational formula for the polynomials Gn1 ,n21 2 (x; k; p) in terms of Kn1 ,1n2 2 (x; k; p). In Section 4, the linear generating functions and a recurrence formula are obtained. In Section 5, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships for the extended multiple Hermite polynomials (α,α ,α ) Gn1 ,n21 2 (x; k; p). In the last section, some applications are given. 2. Explicit expression of the polynomials Chatterjea [4] defined the generalized Laguerre polynomials via the Rodrigues formula by 1 −α pxk n α+n −pxk x e D (x e ); p, k ∈ N, n! and using the following generalized rule for the derivative of a sufficiently differentiable function of x [15], (α)
Tkn (x, p) =
D (f (x)) = s
s − (−1)k k=0
where Dn = (α)
dn , dxn
k!
k − k j D f ( x) (−1) xk−j Ds xj , k
j
j =0
(4)
(5)
he obtained the explicit formula of the polynomials as
Tkn (x, p) =
n − pi i=0
i!
i − i α + n + kj j x (−1) . ki
j
j=0
n
(6)
Furthermore, in [4], Chatterjea derived the formula Dn (xα+n e−px Y ) = n!xα e−px k
k
n − xr
r! r =0
(α+r )
Tk(n−r ) (x, p)Dr Y
where Y is any sufficiently differentiable function of x.
(7)
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C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
Now, using (3), (6) and (7), we get the following k k 1 ,α2 ) (x; k; p) = x−α epx e−α1 x Dn1 eα1 x e−α2 x Dn2 xα e−px eα2 x G(α,α n 1 ,n 2
= n2 !x−α epx e−α1 x Dn1 eα1 x e−α2 x xα−n2 e−px k
k
n2 − xr
r! r =0
(α−n +r )
Tk(n −2r ) (x, p)Dr (eα2 x ) 2
n2 n2 −r i − i − 1 − p i α + kj k (−1)j α2r Dn1 (xα+r +ki−n2 e−px eα1 x ) r ! i ! j n − r 2 r =0 i =0 j =0 n− n2 i 2 −r i − − 1 p i α + kj j = n1 ! n2 ! (−1) α2r r ! i =0 i ! j =0 j n2 − r r =0 k
= n2 !x−α epx e−α1 x
× x−n2 −n1 +r +ki
n1 − xl l=0
l!
(α−n +r +ki−n1 +l)
Tk(n −2l) 1
(x, p)α1l
− n2 n2 −r i − n1 i − 1 − 1 p i α + kj j = n1 ! n2 ! (−1) α2r r ! i ! j n − r l ! 2 r =0 l =0 i =0 j =0 n l1 1 −l l 1 − − p l1 α − n2 + r + ki + kh × (−1)h α1l x−n2 −n1 +r +l+ki+kl1 . l ! h n − l 1 1 l =0 h=0 1
Then, we can state the following theorem, which gives the explicit form of the polynomials defined by (3). (α,α ,α2 )
(x; k; p) has the explicit expression i 2 −r − − α r xr − α l xl n− pi i α + kj ki−n2 1 2 j 1 ,α2 ) G(α,α (− 1 ) x ( x ; k ; p ) = n ! n ! 1 2 n 1 ,n 2 r ! l=0 l! i=0 j=0 i! j n2 − r r =0
Theorem 2.1. The family of polynomials Gn1 ,n21 n1
n2
n 1 −l
×
l1 −− pl1
l1 =0 h=0
l1 !
(−1)h
l1 h
α − n2 + r + +ki + kh kl1 −n1 x . n1 − l
(8)
Comparing (2) and (8), it can be stated that: (α,α ,α2 )
Corollary 2.2. The family of polynomials Gn1 ,n21 1 ,α2 ) G(α,α (x; k; p) = n 1 ,n 2
n2
−
(x; k; p) has the explicit expression
n1
α2r xr
−
r =0
α1l xl (−1)n1 −l+n2 −r
l=0
(0,α ,α2 )
n n 1 2 l
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
r
(x; k; p)x−n1 −n2 .
(δ,α ,α )
x; 2; − 2δ = Hn1 ,n21 2 (x), then as a consequence of Corollary 2.2 we (α ,α ) have the following relation between the multiple Hermite polynomials and the polynomials Kn1 ,1n2 2 (x; k; p): Remark 2.1. It should be noticed that, since Gn1 ,n12
1 ,α2 ) Hn(δ,α (x) = 1 ,n2
n2 −
α2r xr
r =0
n1 −
α1l xl (−1)n1 −l+n2 −r
n n 1 2
l =0
l
r
(−n +r −n +l,−n2 +r )
K(n −2l,n −r1) 1 2
x; 2; −
δ 2
x−n1 −n2 .
3. Operational formulas In [4], Chatterjea obtained the operational formula x−α epx Dn (xα+n e−px Y ) = k
k
n ∏
(xD − pkxk + α + j)Y
(9)
j =1
where Y is any sufficiently differentiable function of x. Now use Leibniz’ formula to find dn 1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y dx 1 dx 2 k
= x−α epx e−α1 x
dn1 dxn1
eα1 x e−α2 x
n2 − n2 s=0
s
Dn2 −s (eα2 x )Ds (xα e−px Y ) k
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
4881
then with the help of Eq. (9), we get dn 2 dn1 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y 1 2 dx dx dn 1
k
= x−α epx e−α1 x k
= x−α epx e−α1 x
dxn1
eα1 x e−α2 x
=x
e
e
n2 − n2
k
n2 − n2
s
×
dn 1
α
n −s
s ∏
(xD − pkxk + α − s + j)Y
j =1
n1 − n1
l
n1 − n1
l
l =0
k
n −s
k
s ∏
(xD − pkxk + α − s + j)Y
j =1
l=0
α2 2
α2 2 eα2 x xα−s e−px
eα1 x xα−s e−px
dxn1
n 2 −s 2
s
n2 − n2 s=0
l ∏
n −s
α2 2
s
s=0
= x−α epx
s
s=0
s=0
−α pxk −α1 x
n2 − n2
(xD − pkxk + α − s − l + i)
D
n1 −l
(e
α1 x
)D
l
x
α−s −pxk
e
s ∏
(xD − pkx + α − s + j)Y k
j=1
α1 1 xα−s−l e−px n −l
k
s ∏ (xD − pkxk + α − s + j)Y .
i =1
j =1
Hence, we obtain the first equality k
x−α epx e−α1 x
=
dn1 dxn1
n2 − n2 s=0
s
eα1 x e−α2 x
n −s
α2 2
dn 2
n1 − n1
l
l=0
xα e−px eα2 x Y k
dxn2
n −l
α1 1 x−s−l
l ∏
(xD − pkxk + α − s − l + i)
i =1
s ∏
(xD − pkxk + α − s + j)Y .
j =1
Now, taking (7) into consideration, we have dn1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y 1 2 dx dx k
= n2 !x−α epx e−α1 x k
= n2 !x−α epx e−α1 x n 2 −r
×
dn1 dxn1 dn1 dxn1
eα1 x e−α2 x xα−n2 e−px
k
eα1 x e−α2 x xα−n2 e−px
k
i − pi − xki (−1)j i! i=0 j =0
i
α + kj
j
n2 − r
n2 − xr
r! r =0
(α−n +r )
Tk(n −2r ) (x, p)Dr (eα2 x Y ) 2
n2 − xr
r! r =0 − r r
r 1 =0
r1
r −r1 α2 x
α2
e
Dr1 (Y )
r n2 n2 −r i − i − 1 − p i α + kj − r k r −r (−1)j α2 1 Dn1 (xα−n2 +r +ki e−px eα1 x Dr1 (Y )) r ! i =0 i ! j =0 j n2 − r r = 0 r 1 r =0 1 n n − r i 2 2 i − − − p i α + kj 1 k = n1 !n2 !x−α epx e−α1 x (−1)j r ! i=0 i! j=0 j n2 − r r =0 n r 1 − − r xl (α−n2 +r +ki−n1 +l) k r −r × α2 1 xα−n2 +r +ki−n1 e−px Tk(n −l) (x, p)Dl (eα1 x Dr1 (Y )) 1 l ! r 1 r 1 =0 l =0 r n− n2 i 2 −r i − − 1 p i α + kj − r k r −r j −α pxk −α1 x = n1 !n2 !x e e (−1) α2 1 xα−n2 +r +ki−n1 e−px r ! i ! j n − r r 2 1 i=0 j =0 r 1 =0 r =0 n n − l l l 1 1 1 − xl − pl1 kl1 − l1 α − n2 + r + ki + kh − l × x (−1)h Dl−h1 (eα1 x )Dh1 (Dr1 (Y )) l ! l ! h n − l h 1 1 1 l =0 l 1 =0 h=0 h1 =0 n− n2 n1 l1 i l l n− r r − 2 −r 1 −l l1 i − − α2 x α1 x p ki i α + kj p kl1 − l1 j h = n2 !n1 ! x (−1) x (−1) r ! l ! i ! j n − r l ! h 2 h=0 r =0 l=0 i=0 j =0 l1 =0 1 r l α − n2 + r + ki + kh −n1 −n2 − r 1 − l 1 h 1 +r 1 × x D (Y ). r1 n1 − l r1 α2 h =0 h1 α h1 r1 =0 1 1 k
= n2 !x−α epx e−α1 x
(10)
4882
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Hence, the second equality will be dn 1 dn 2 k k x−α epx e−α1 x n eα1 x e−α2 x n xα e−px eα2 x Y dx 1 dx 2
=
n2 −
α2r xr
n1 −
α1l xl (−1)n1 −l+n2 −r
n n 1 2 l
l =0
r =0
r
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
(x, k, p)
l r − l 1 h 1 +r 1 r 1 − (Y )x−n1 −n2 . D × r1 r1 α2 h =0 h1 α h1 r1 =0 1 1
(11)
Thus, it follows from (10) and (11). (α ,α )
Theorem 3.1. Let Kn1 ,1n2 2 (x; k; p) be defined by (2). Then we have n2 − n2
r
r =0
=
n2 − r =0
×
n −r
α2 2
n1 − n1
l
l =0
α2r xr
n1 −
n −l
α1 1 x−r −l
l ∏
(xD − pkxk + α − r − l + i)
i=1
α1l xl (−1)n1 −l+n2 −r
(xD − pkxk + α − r + j)Y
j =1
n n 1 2 l
l =0
r ∏
r
(α−n +r −n +l,α−n2 +r )
K(n −l,2n −r ) 1 1 2
(x; k; p)
r l − r 1 − l 1 h 1 +r 1 D (Y )x−n1 −n2 . r1 h1 r h α 1 1 α 2 h 1 =0 r1 =0 1
(12)
Setting Y = 1 in (12), we obtain the following theorem. (α,α ,α2 )
Theorem 3.2. For Gn1 ,n21
1 ,α2 ) G(α,α (x; k; p) = n 1 ,n 2
(x; k; p), we have
n2 − n2
r
r =0
n −r
α2 2
n1 − n1
l
l =0
n −l
α1 1 x−r −l
l
×
∏
(xD − pkxk + α − r − l + i)
r ∏
i=1
(xD − pkxk + α − r + j).
(13)
j=1
Furthermore, letting k = 2, p = − 2δ and α = 0 in (13), we get the following result. (δ,α ,α2 )
Corollary 3.3. For the multiple Hermite polynomials Hn1 ,n21 1 ,α2 ) Hn(δ,α (x) = 1 ,n2
n2 − n2 r =0
r
n −r
α2 2
n1 − n1 l =0
l
n −l
α1 1 x−r −l
(x), we have
l ∏
(xD + δ x2 − r − l + i)
i=1
r ∏
(xD + δ x2 − r + j).
j=1
Moreover, for each suitable choice of the sufficiently differentiable function Y , Theorem 3.1 can be shown to yield various interesting operational formulas for the polynomials defined by (3) and of course for the multiple Hermite polynomials. 4. Generating functions and recurrence relations In this section we obtain the generating functions and a recurrence relation for a class of polynomials defined by (3). The first result of this section is the following theorem: (α,α ,α2 )
Theorem 4.1. Let Gn1 ,n21 G∗ (x; t1 ; t2 ) =
(x; k; p) be the polynomials defined by (3). Then we have
∞ − n 1 ,n 2 = 0
n
1 ,α2 ) Gn(α,α (x; k; p) 1 ,n 2
n
t2 2 t1 1 n2 ! n1 !
= x−α (x + t1 + t2 )α eα1 t1 +α2 t2 +px Proof. Direct computations yield that
k −p(x+t
k 1 +t2 )
.
(14)
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
∞ − n1 ,n2 =0
=x
1 ,α2 ) G(α,α (x; k; p) n1 ,n2
−α pxk −α1 x
e
e
=x
e
e
k
= x−α epx e−α1 x k
k
= x−α epx e−α1 x k
= x−α epx e−α1 x
dxn1
∞ − dn1 n1 =0
= x−α epx e−α1 x
dxn1
∞ − dn1 n1 =0
4883
n t1 1
n2 ! n1 !
∞ − dn1 n1 =0
−α pxk −α1 x
n t2 2
dxn1
e
α1 x −α2 x
e
∞ − dn 2 n 2 =0
e
α1 x −α2 x
e
1
dxn2
eα1 x e−α2 x
1
x e
n2 !
z−x
C1
zα e
2π i
e
C1
−pz k
eα 2 z
z − x − t2
n
t1 1 n1 !
k ∞ z α e−pz eα2 z −
2π i
n2
α −pxk α2 x t2
dz
dz
z−x
n 2 =0
n2
t2
n
t1 1 n1 !
n
t1 1 n1 !
n1
∞ − dn1
t k eα1 x (x + t2 )α e−p(x+t2 ) eα2 t2 1 n1 dx n 1! n1 =0 k ∞ eα1 η (η + t2 )α e−p(η+t2 ) eα2 t2 − t1
1 2π i
2π i
n 1 =0
k eα1 η (η + t2 )α e−p(η+t2 ) eα2 t2
1
η−x
C2
η − x − t1
C2
n1
η−x
dη
dη
where C1 is a circle in the complex z-plane, cut along the negative real axis, (centered at z = x + t2 ) with radius ε > 0, which is described in the positive direction (counter-clockwise) and C2 is a circle (centered at η = x + t1 ) in the complex
t1 η-plane, cut along the negative real axis, with sufficiently small radius and zt−2 x < 1, η− < 1. Thus x n
∞ − n1 ,n2 =0
1 ,α2 ) G(α,α (x; k; p) n1 ,n2
n
t2 2 t1 1 n2 ! n1 !
= x−α (x + t1 + t2 )α eα1 t1 +α2 t2 +px
k −p(x+t
1 + t2 )
k
.
Remark 4.1. Letting k = 2, p = − 2δ and α = 0 in (14), we get the generating function for the multiple Hermite polynomials which was already obtained in [2, Eq (6)]. (α,α ,α2 )
Theorem 4.2. For Gn1 ,n21
(x; k; p), we have
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) = 1 ,n2
n1 − n2 − n1 n2 l 1 =0 l 2 =0
l1
l2
(α,α ,α )
Gn1 −l11 ,n22 −l2 x; k;
p (β,β1 ,β2 ) p Gl1 ,l2 . x; k; 2 2
Proof. Taking (14) into consideration, we have ∞ − ∞ −
n
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) 1 ,n 2
n 1 =0 n 2 =0
n
t2 2 t1 1 n2 ! n1 !
= x−(α+β) (x + t1 + t2 )(α+β) e(α1 +β1 )t1 +(α2 +β2 )t2 +px −p(x+t1 +t2 ) n +l n +l ∞ ∞ − − p (β,β1 ,β2 ) p t1 1 1 t2 2 2 1 ,α2 ) = G x ; k ; . Gn(α,α x ; k ; , n l , l 1 2 1 2 2 2 n1 !l1 !n2 !l2 ! n ,n = 0 l ,l = 0 k
1
2
k
1 2
Taking n1 −→ n1 − l1 and n2 −→ n2 − l2 , we get ∞ − ∞ −
n
1 +β1 ,α2 +β2 ) Gn(α+β,α (x; k; p) 1 ,n 2
n 1 =0 n 2 =0
=
n1 − n2 ∞ − − n1 ,n2 =0 l1 =0 l2 =0
Hence the result.
(α,α ,α )
n
t2 2 t1 1 n2 ! n1 !
Gn1 −l11 ,n22 −l2 x; k;
n
n
p (β,β1 ,β2 ) p t1 1 t2 2 Gl1 ,l2 x; k; . 2 2 (n1 − l1 )!l1 !(n2 − l2 )!l2 !
Letting k = 2, p = − 2δ and α = 0 in (15), we get the following recurrence relation.
(15)
4884
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
Corollary 4.3. The recurrence relation for the multiple Hermite polynomials is 1 +β1 ,α2 +β2 ) Hn(δ,α (x) ,m
=
n − m − n m
k
k=0 l=0
l
δ ,α ,α 1 2
δ ,β ,β 1 2
Hn−k,m−l (x)Hk,l 2
2
(x).
5. Multilateral generating functions Very recently several substantially more general families of bilinear, bilateral, and mixed multilateral finite-series relationships and generating functions for the multiple Laguerre and multiple Hermite polynomials are given in [16]. In this section, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships for the extended (α,α ,α ) multiple Hermite polynomials Gn1 ,n21 2 (x; k; p) by applying the same method used in [17,16]. We start with the following theorem. Theorem 5.1. Corresponding to an identically nonvanishing function Φµ(1) (ξ1 , ξ2 , . . . , ξs ) of s (real or complex) variables ξ1 , ξ2 , . . . , ξs (sϵ N = N0 \ {0}) and of (complex) order µ, let
∆(1) (ξ1 , ξ2 , . . . , ξs ; w) :=
∞ −
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )w l
al ̸= 0.
(16)
l =0
Suppose also that
n
k
(1) N2 N1 − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )
Θn,N1 ,N2 (x; ξ1 , ξ2 , . . . , ξs ; ζ , ρ) :=
k=0
l =0
(n − N2 k)!(k − N1 l)!
(α+λl,α +λ l,α2 +λ2 l)
× Gn−N2 k,k1−N11l
(x; k; p)ζ k ρ l (nϵ N)
(17)
then ∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
=∆
(1)
ξ1 , ξ2 , . . . , ξs ;
t2
,
N t1 2
γ
t1n
N t2 1
γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 )
xλ
G∗ (x; t1 ; t2 )
(18)
provided that each member of (18) exists. The notation
n q
means that the greatest integer is less than or equal to nq .
Proof. Considering (17) into (18), we get
∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
t2 N
,
γ
t1n =
N
t1 2 t2 1
n
k
(1) N2 N1 ∞ − − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) n=0 k=0
l=0
(n − N2 k)!(k − N1 l)! k
(α+λl,α +λ l,α +λ l) × Gn−N2 k,k1−N11l 2 2 (x; k; p)
t2
γ
N
t2 1
t1 2
N
l t1n .
Using n −→ n + N2 k
∞ −
x; ξ1 , ξ2 , . . . , ξs ;
Θn,N1 ,N2
n =0
t2 N
,
γ
=
N
t1 2 t2 1
k
(1) N1 ∞ − ∞ − − al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) n=0 k=0 l=0
n!(k − N1 l)!
(α+λl,α +λ l,α +λ l) k−N l × Gn,k−N1 l1 1 2 2 (x; k; p)γ l t1n t2 1 .
Taking k −→ k + N1 l and using (14), we find ∞ − n =0
Θn,N1 ,N2
x; ξ1 , ξ2 , . . . , ξs ;
t2 N
,
γ N
t1 2 t2 1
=
∞ −
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )γ l x−(α+λl) (x + t1 + t2 )α+λl
l =0
× exp((α1 + λ1 l)t1 + (α2 + λ2 l)t2 + pxk − p(x + t1 + t2 )k )
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
∞ −
=
(1)
al Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs )
γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 )
4885
l
xλ
l=0
× x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) G∗ (x; t1 ; t2 ). = ∆(1) ξ1 , ξ2 , . . . , ξs ; λ x
Theorem 5.2. Corresponding to an identically nonvanishing function Φµ(21),µ2 (ξ1 , ξ2 , . . . , ξs ) of s (real or complex) variables ξ1 , ξ2 , . . . , ξs (sϵ N = N0 \ {0}) and of (complex) order µ, let (2)
∆Ψ1 ,Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; τ1 , τ2 ) :=
∞ − ∞ −
(2)
k
k
ak1 ,k2 ̸= 0.
ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )τ1 1 τ2 2
k1 =0 k2 =0
Suppose also that
n1
n2
(2) q1 q2 − − ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
1 ,λ2 ,Ψ1 ,Ψ2 Λnµ,λ (x; ξ1 , ξ2 , . . . , ξs ; ζ1 , ζ2 ) := 1 ,n 2 ,q 1 ,q 2
(n1 − q1 k1 )!(n2 − q2 k2 )!
k1 =0 k2 =0
(α,α +λ k ,α +λ k2 )
× Gn1 −q11 k1 ,1n21−q22 k2 2
k
k
(x; k; p)ζ1 1 ζ2 2
(19)
then ∞ − n1 ,n2 =0
v1 v2 n n 1 ,λ2 ,Ψ1 ,Ψ2 , t1 1 t2 2 Λµ,λ x ; ξ , ξ , . . . , ξ ; 1 2 s q1 q2 n1 ,n2 ,q1 ,q2 t1
t2
= ∆(Ψ21),Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; v1 eλ1 t1 , v2 eλ2 t2 )G∗ (x; t1 ; t2 ).
(20)
Proof. Considering (19) into (20), we get
∞ −
n1 ,n2 =0
1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x; ξ1 , ξ2 , . . . , ξs ; n1 ,n2 ,q1 ,q2
v1 q
,
v2
n
n
t1 1 t2 2 =
q
t1 1 t2 2
n1
n2
(2) q1 q2 ∞ − − − ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
(n1 − q1 k1 )!(n2 − q2 k2 )! k1 k2 v1 v2 (α,α +λ k ,α +λ k ) n n × Gn1 −q11 k1 ,1n21−q22 k2 2 2 (x; k; p) q1 t1 1 t2 2 . q2 n1 ,n2 =0 k1 =0 k2 =0
t1
t2
Taking n2 −→ n2 + q2 k2 ∞ −
n1 ,n2 =0
1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x; ξ1 , ξ2 , . . . , ξs ; n1 ,n2 ,q1 ,q2
v1 q t1 1
,
v2
n
n
t1 1 t2 2 =
q t2 2
∞ −
n1 q1
(2) ∞ a −− k1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )
n1 ,n2 =0 k1 =0 k2 =0
(n1 − q1 k1 )!n2 !
(α,α +λ k ,α +λ k ) k k n −q k n × Gn1 −q11 k1 ,1n21 2 2 2 (x; k; p)v11 v22 t1 1 1 1 t2 2 .
Taking n1 −→ n1 + q1 k1 and considering (14), we have ∞ − n1 ,n2 =0
=
v1 v2 n n 1 ,λ2 ,Ψ1 ,Ψ2 Λµ,λ x ; ξ , ξ , . . . , ξ ; , t1 1 t2 2 1 2 s q1 q2 n1 ,n2 ,q1 ,q2
∞ − ∞ − k1 =0 k2 =0
=
∞ − ∞ −
t1
∞ − n1 ,n2 =0
t2
n1
t 1 +λ1 k1 ,α2 +λ2 k2 ) Gn(α,α (x; k; p) 1 1 ,n 2
n
t2 2
n1 ! n2 !
(2)
x−α (x + t1 + t2 )α exp((α1 + λ1 k1 )t1 + (α2 + λ2 k2 )t2 + pxk − p(x + t1 + t2 )k )
k1 =0 k2 =0
× ak1 ,k2 Φµ(21)+Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )v11 v22 ∞ − ∞ − (2) = ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )(v1 eλ1 t1 )k1 (v2 eλ2 t2 )k2 k
k
k
ak1 ,k2 Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs )v11 v22
k
k1 =0 k2 =0
× x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) = ∆(Ψ21),Ψ2 ,µ1 ,µ2 (ξ1 , ξ2 , . . . , ξs ; v1 eλ1 t1 , v2 eλ2 t2 )G∗ (x; t1 ; t2 ).
4886
C. Kaanoğlu, M. Ali Özarslan / Journal of Computational and Applied Mathematics 235 (2011) 4878–4887
6. Applications of main results (1)
(2)
If the multivariable functions Φµ+Ψ l (ξ1 , ξ2 , . . . , ξs ) and Φµ1 +Ψ1 k1 ,µ2 +Ψ2 k2 (ξ1 , ξ2 , . . . , ξs ) can be selected by means of several functions of one variable then further applications of Theorems 5.1 and 5.2 can be given. It should be noted here that, one should use Theorem 5.1 in order to give bilinear, bilateral, mixed multilateral finite-series relationships and generating functions between the polynomials of which contain one summation symbol in its generating relation and the extended multiple Hermite polynomials. On the other hand, one cannot give any bilinear, bilateral, mixed multilateral finite-series relationships and generating functions between the polynomials of which contain more than one summation symbol in its generating relation and the extended multiple Hermite polynomials by using Theorem 5.1. But one can use Theorem 5.2 to achieve this problem. In this section we give the following illustrative examples. (1) Example 6.1. Letting s = 1, µ = 0, Ψ = 1, ξ1 = x, al = 1 and Φl (x) = Lαn (x), where Lαn (x) is the classical Laguerre polynomial (see [12]) and it is known that ∞ −
Lαn (x)t n = (1 − t )−α−1 exp(x(1 − (1 − t )−1 ))
n =0
then by Theorem 5.1, we obtain the generating function between the classical Laguerre polynomials and the extended multiple Hermite polynomials
n
k
(α+λl,α +λ l,α +λ l) N2 N1 ∞ − − − Lαn (x)Gn−N2 k,k1−N11l 2 2 (x; k; p) n=0 k=0
(n − N2 k)!(k − N1 l)!
l =0
t2 N
t1 2
k
γ N
t2 1
l t1n
= x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) −α−1 −1 γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) γ (x + t1 + t2 )λ exp(λ1 t1 + λ2 t2 ) . × 1− exp x 1 − 1 − λ λ x
x
Choosing s = 1, ξ1 = x, µ1 = µ2 = 0, Ψ1 = Ψ2 = 1, ak1 ,k2 = Laguerre I polynomial (see [15]), in Theorem 5.2, we get k
1 k1 !k2 !
∞ − ∞ −
(α ,α ;β)
Lk1 1,k2 2
(α ,α ;β)
(x), where Lk1 1,k2 2
(x) is a
k
τ1 1 τ2 2 k1 ! k2 ! k1 =0 k2 =0 1 β(t1 + t2 − t1 t2 ) exp = x (1 − t1 )α1 (1 − t2 )α2 (1 − t1 )(1 − t2 )
(2)
∆1,1,0,0 (x; τ1 , τ2 ) :=
(α ,α ;β)
(2)
and Φk1 ,k2 (x) = Lk1 1,k2 2
(x)
for |t1 | < 1, |t2 | < 1
and therefore, the generating relation between Laguerre I polynomials and extended multiple Hermite polynomials is ∞ −
n1 q1
n2 q2
1 +λ1 k1 ,α2 +λ2 k2 ) − − Lk(α1,k,α2 ;β) (x)Gn(α,α (x; k; p) v1 k1 v2 k2 1 2 1 −q1 k1 ,n2 −q2 k2
n1 ,n2 =0 k1 =0 k2 =0
(n1 − q1 k1 )!(n2 − q2 k2 )!k1 !k2 !
q
t1 1
q
t2 2
n
n
t1 1 t2 2 1
= x−α (x + t1 + t2 )α exp(α1 t1 + α2 t2 + pxk − p(x + t1 + t2 )k ) (1 − v1 eλ1 t1 )α1 (1 − v2 eλ2 t2 )α2 λ1 t1 λ2 t2 λ1 t1 +λ2 t2 β(v1 e + v2 e − v1 v2 e ) x for v1 eλ1 t1 < 1, v2 eλ2 t2 < 1. × exp λ t λ t 1 1 2 2 (1 − v1 e )(1 − v2 e ) Acknowledgements The authors are very grateful to the anonymous referees for the careful reading of the paper. References [1] E.D. Rainville, Special Functions, The Macmillian Company, New York, 1960. [2] D.W. Lee, Properties of multiple Hermite and multiple Laguerre polynomials by the generating function, Integral Transforms Spec. Funct. 18 (2007) 855–869. [3] S.K. Chatterjea, A generalization of Laguerre polynomials, Collect. Math. 15 (1963) 285–292. [4] S.K. Chatterjea, On a generalization of Laguerre polynomials, Rend. Semin. Mat. Univ. Padova 34 (1964) 180–190. [5] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalization of Hermite polynomials, Duke Math. J. 29 (1962) 51–64.
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