Proyecciones Journal of Mathematics Vol. 30, No 2, pp. 213-245, August 2011. Universidad Cat´olica del Norte Antofagasta - Chile
Triple generating functions of Jacobi polynomials of two variables MUMTAZ AHMAD KHAN ALIGARH MUSLIM UNIVERSITY, INDIA ABDUL HAKIM KHAN ALIGARH MUSLIM UNIVERSITY, INDIA SAYED MOHAMMAD ABBAS ALIGARH MUSLIM UNIVERSITY, INDIA Received : March 2011. Accepted : May 2011
Abstract The present paper is a study of triple generating functions of Jacobi polynomials of two variables.
Keywords : Triple generating functions.
A. M. S. Subject Classification : 33C45.
214
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
1. INTRODUCTION In our paper [6] we discussed some generating and double generating functions of Jacobi polynomials of two variables. Now in this paper we obtain some triple generating functions of Jacobi polynomials of two variables. In our investigation we require the following results: Jacobi polynomials of two variables are defined by [10] Pn(α1 , β1 ; α2 , β2 ) (x, y) =
(1 + α1 )n (1 + α2 )n (n!)2
(1.1) ∙ ¸ 1−y 1−x , ×F2 −n, 1 + α2 + β2 + n, 1 + α1 + β1 + n; 1 + α2 , 1 + α1 ; . 2 2 The confluent hypergeometric functions of two variables are defined by [9] ∞ X ∞ X (α)m+n (β)m xm yn
φ1 [α, β; γ; x, y] =
(1.2)
(γ)m+n
m=0 n=0
£
¤
φ2 β, β 0 ; γ; x, y =
(1.3)
(1.4)
∞ X ∞ X (β)m (β 0 )n xm y n ∞ X ∞ X (β)m xm yn
m=0 n=0
£
¤
(1.5) Ψ1 α, β; γ, γ 0 ; x, y =
£
¤
(γ)m+n m! n!
m=0 n=0
φ3 [β; γ; x, y] =
(γ)m+n m! n!
, |x| < 1, |y| < ∞ ,
, |x| < ∞,
(γ)m (γ 0 )n m! n!
∞ X ∞ X (α)m+n xm y n
m=0 n=0
|y| < ∞ ,
, |x| < ∞, |y| < ∞ ,
∞ X ∞ X (α)m+n (β)m xm y n
m=0 n=0
(1.6) Ψ2 α; γ, γ 0 ; x, y =
m! n!
(γ)m (γ 0 )n m! n!
, |x| < 1, |y| < ∞ ,
, |x| < ∞,
|y| < ∞ ,
(1.7) £
¤
Ξ1 α, α0 , β; γ; x, y =
∞ ∞ X X (α)m (α0 )n (β)m xm y n
m=0 n=0
(γ)m+n
m! n!
, |x| < 1, |y| < ∞.
Triple generating functions of Jacobi polynomials of two variables 215
2. TRIPLE GENERATING FUNCTIONS (α1 , β1 ; α2 , β2 )
The polynomials Pn functions:
(x, y) admits the following triple generating
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (λ)k (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 )n (1 + α2 )k+n
×Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk t
−1−α1 −β1
= e (1 − u) (2.1)
× Φ1
1 F1
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
∙
#
¸
1 1 + α2 + β2 , λ; 1 + α2 ; v, (y − 1)t , 2
∞ X ∞ ∞ X X n!(λ)m (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 )m+n (1 + α2 )n
×Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um vk t
−1−α2 −β2
= e (1 − v)
1 F1
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
#
¸
1 × Φ1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t , 2
(2.2)
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m (λ)k m!k!(1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k t
−1−α1 −β1
= e (1 − u) (2.3)
1 F1
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
∙
¸
1 × Φ2 λ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)m (1 + α2 + β2 )k m!k!(1 + α1 )m+n (1 + α2 )n m=0 k=0 n=0
#
216
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k t
−1−α2 −β2
= e (1 − v)
(2.4)
1 F1
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
#
¸
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t , 2
× Φ2
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (λ)k (1 + α2 + β2 )k
m!k!(µ)k (1 + α1 )n (1 + α2 )n
m=0 k=0 n=0
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk = et (1 − u)−1−α1 −β1 1 F1
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
∙
#
¸
1 × Ψ1 1 + α2 + β2 , λ; µ, 1 + α2 ; v, (y − 1)t , 2
(2.5)
∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(µ)m (1 + α1 )n (1 + α2 )n
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk t
−1−α2 −β2
= e (1 − v)
(2.6)
× Ψ1
1 F1
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
#
¸
1 1 + α1 + β1 , λ; µ, 1 + α1 ; u, (x − 1)t , 2
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(λ)k (1 + α1 )n (1 + α2 )n
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk t
−1−α1 −β1
= e (1 − u)
1 F1
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
#
Triple generating functions of Jacobi polynomials of two variables 217
(2.7)
× Ψ2
∙
¸
1 1 + α2 + β2 ; λ, 1 + α2 ; v, (y − 1)t , 2
∞ X ∞ ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m!k!(λ)m (1 + α1 )n (1 + α2 )n
m=0 k=0 n=0
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk t
−1−α2 −β2
= e (1 − v) (2.8)
1 F1
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
¸
1 × Ψ2 1 + α1 + β1 ; λ, 1 + α1 ; u, (x − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k (1 + α2 + β2 )k
m!k!(1 + α1 )m+n (1 + α2 )k+n
m=0 k=0 n=0
×Pn(α1 +m, β1 −n; α2 +k, β2 −n) (x, y)tn um vk t
= e Φ1
(2.9)
∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Φ1 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)m (µ)k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k t
= e Φ2
(2.10)
∙
∙
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
¸
¸
1 × Φ2 µ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (µ)k (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(δ)m (ν)k (1 + α1 )n (1 + α2 )n
#
218
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas ×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk ∙
1 = et Ψ1 1 + α1 + β1 , λ; δ, 1 + α1 ; u, (x − 1)t 2 (2.11)
× Ψ1
∙
¸
¸
1 1 + α2 + β2 , µ; ν, 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(λ)m (µ)k (1 + α1 )n (1 + α2 )n
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk t
= e Ψ2
(2.12)
∙
1 1 + α1 + β1 ; λ, 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Ψ2 1 + α2 + β2 ; µ, 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k
m=0 k=0 n=0
m!k!(1 + α1 )m+n (1 + α2 )k+n
×Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um v k t
= e Φ1
(2.13)
× Φ2
∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 µ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (µ)k (1 + α2 + β2 )k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −n) (x, y)tn um v k ∙
1 = et Φ2 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2 (2.14)
× Φ1
∙
¸
¸
1 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 219
∞ X ∞ ∞ X X
n!(λ)m (1 + α1 + β1 )m m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um v k t
= e Φ1
(2.15)
∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Φ3 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2 ∞ ∞ X ∞ X X
n!(λ)k (1 + α2 + β2 )k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −n) (x, y)tn um v k ∙
= e Φ3
(2.16)
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
t
∙
¸
1 × Φ1 1 + α2 + β2 , λ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)m m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k = e Φ2
∙
× Φ3
∙
t
(2.17)
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
¸
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
∞ ∞ X ∞ X X
n!(λ)k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k
220
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas ∙
= e Φ3
(2.18)
× Φ2
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
t
∙
¸
1 λ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k (1 + α2 + β2 )k
m!k!(δ)k (1 + α1 )m+n (1 + α2 )n
m=0 k=0 n=0
×Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um vk ∙
1 = et Φ1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2 (2.19)
∙
¸
¸
1 × Ψ1 1 + α2 + β2 , µ; δ, 1 + α2 ; v, (y − 1)t , 2 ∞ X ∞ ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k (1 + α2 + β2 )k
m!k!(δ)m (1 + α1 )n (1 + α2 )k+n
m=0 k=0 n=0
×Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk t
= e Ψ1
(2.20)
∙
1 1 + α1 + β1 , λ; δ, 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Φ1 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 )m+n (µ)k (1 + α2 )n
×Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um vk t
= e Φ1
(2.21)
× Ψ2
∙
∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2 ¸
¸
1 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 221
∞ X ∞ ∞ X X n!(1 + α1 + β1 )m (λ)k (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(µ)m (1 + α1 )n (1 + α2 )k+n
×Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk ∙
= e Ψ2
(2.22)
¸
1 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
t
∙
¸
1 × Φ1 1 + α2 + β2 , λ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k (ν)k
m=0 k=0 n=0
m!k!(1 + α1 )m+n (1 + α2 )k+n
×Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2
t
= e Φ1
(2.23)
∙
¸
¸
1 × Ξ1 µ, 1 + α2 + β2 , ν; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (µ)m (ν)k (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 )m+n (1 + α2 )k+n
×Pn(α1 +m, β1 −m−n; α2 +k, β2 −n) (x, y)tn um v k = e Ξ1
∙
× Φ1
∙
t
(2.24)
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2 ¸
1 1 + α2 + β2 , ν; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (µ)k (1 + α2 + β2 )k m!k!(1 + α1 )m+n (δ)k (1 + α2 )n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k
¸
222
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas t
= e Φ2
(2.25)
× Ψ1
∙
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 1 + α2 + β2 , µ; δ, 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (1 + α1 + β1 )m (µ)k m!k!(δ) m (1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 = et Ψ1 1 + α1 + β1 , λ; δ, 1 + α1 ; u, (x − 1)t 2 (2.26)
∙
¸
¸
1 × Φ2 µ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2 ∞ X ∞ ∞ X X
n!(λ)m (1 + α2 + β2 )k m!k!(µ)k (1 + α1 )m+n (1 + α2 )n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k t
= e Φ2
(2.27)
∙
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Ψ2 1 + α2 + β2 ; µ, 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m (λ)k m!k!(µ)m (1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k t
= e Ψ2
(2.28)
× Φ2
∙
∙
¸
1 1 + α1 + β1 ; µ, 1 + α1 ; u, (x − 1)t 2 ¸
1 λ, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 223
∞ X ∞ ∞ X X
n!(λ)m (µ)k (ν)k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 λ, 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
t
= e Φ2
(2.29)
∙
¸
¸
1 × Ξ1 µ, 1 + α2 + β2 , ν; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)m (µ)m (ν)k m!k!(1 + α1 )m+n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
t
= e Ξ1
(2.30)
∙
¸
¸
1 × Φ2 ν, 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)k (1 + α2 + β2 )k m!k!(µ)k (1 + α1 )m+n (1 + α2 )n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k t
= e Φ3
(2.31)
× Ψ1
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
∙
¸
1 1 + α2 + β2 , λ; µ, 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (1 + α1 + β1 )m m!k!(µ)m (1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k
224
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas t
= e Ψ1
(2.32)
∙
1 1 + α1 + β1 , λ; µ, 1 + α1 ; u, (x − 1)t 2
× Φ3
∙
¸
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k m!k!(1 + α1 )m+n (λ)k (1 + α2 )n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k ∙
¸
1 = et Φ3 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2 (2.33)
∙
¸
1 × Ψ2 1 + α2 + β2 ; λ, 1 + α2 ; v, (y − 1)t , 2 ∞ X ∞ ∞ X X
n!(1 + α1 + β1 )m m!k!(λ)m (1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 1 + α1 + β1 ; λ, 1 + α1 ; u, (x − 1)t 2
t
= e Ψ2
(2.34)
∙
¸
¸
1 × Φ3 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2 ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(µ)m (δ)k (1 + α1 )n (1 + α2 )n
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk = e Ψ1
∙
× Ψ2
∙
t
(2.35)
1 1 + α1 + β1 , λ; µ, 1 + α1 ; u, (x − 1)t 2 ¸
1 1 + α2 + β2 ; δ, 1 + α2 ; v, (y − 1)t , 2
¸
Triple generating functions of Jacobi polynomials of two variables 225
∞ X ∞ ∞ X X n!(λ)k (1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(µ)m (δ)k (1 + α1 )n (1 + α2 )n
×Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um vk ∙
= e Ψ2
(2.36)
¸
1 1 + α1 + β1 ; µ, 1 + α1 ; u, (x − 1)t 2
t
∙
¸
1 × Ψ1 1 + α2 + β2 , λ; δ, 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (µ)k (δ)k (1 + α1 + β1 )m
m=0 k=0 n=0
m!k!(ν)m (1 + α1 )n (1 + α2 )k+n
×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k ∙
1 1 + α1 + β1 , λ; ν, 1 + α1 ; u, (x − 1)t 2
t
= e Ψ1
(2.37)
∙
¸
¸
1 × Ξ1 µ, 1 + α2 + β2 , δ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X n!(λ)m (µ)m (δ)k (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(ν)k (1 + α1 )m+n (1 + α2 )n
×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k ∙
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
t
= e Ξ1
(2.38)
× Ψ1
∙
¸
¸
1 1 + α2 + β2 , δ; ν, 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m (λ)k (µ)k m!k!(ν)m (1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 ×Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um v k
226
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas ∙
= e Ψ2
(2.39)
× Ξ1
¸
1 1 + α1 + β1 ; ν, 1 + α1 ; u, (x − 1)t 2
t
∙
¸
1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (µ)m (1 + α2 + β2 )k m!k!(ν)k (1 + α1 )m+n (1 + α2 )n m=0 k=0 n=0 ×Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k t
= e Ξ1
∙
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Ψ2 1 + α2 + β2 ; ν, 1 + α2 ; v, (y − 1)t . 2
(2.40)
PROOF OF (2.1) ∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (λ)k (1 + α2 + β2 )k
m!k!(1 + α1 )n (1 + α2 )k+n
m=0 k=0 n=0
×Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk =
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (λ)k (1 + α2 + β2 )k
m=0 k=0 n=0
××
m!k!(1 + α1 )n (1 + α2 + k)n (1 + α2 )k
(1 + α1 )n (1 + α2 + k)n (n!)2
n n−r X X (−n)r+s (1 + α1 + β1 + m)r (1 + α2 + β2 + k)s
r!s!(1 + α1 )r (1 + α2 + k)s
r=0 s=0
µ
=
×
1−x 2
¶r µ
1−y 2
¶s
tn um v k
∞ X ∞ X ∞ X n n−r X X (−1)r+s n!(1 + α1 + β1 )m+r (1 + α2 + β2 )k+s (λ)k
m!k!n!r!s!(n − r − s)!(1 + α1 )r (1 + α2 )k+s
m=0 k=0 n=0 r=0 s=0
µ
1−x × 2
¶r µ
1−y 2
¶s
tn um v k
Triple generating functions of Jacobi polynomials of two variables 227
∞ X ∞ X ∞ X ∞ ∞ X X (1 + α1 + β1 )m+r (1 + α2 + β2 )k+s (λ)k
=
m!k!n!r!s!(1 + α1 )r (1 + α2 )k+s
m=0 k=0 n=0 r=0 s=0
×
³
x−1 2
t
=e
´r ³
´ y−1 s n+r+s m k t u v 2
µ ∞ X (1 + α1 + β1 )r 1
r!(1 + α1 )r
2
∞ P ∞ P (1+α2 +β2 )k+s (λ)k
× vk
r=0
k=0 s=0
k!s!(1+α2 )k+s
t
¶r X ∞ (1 + α1 + β1 + r)m m (x − 1)t u ³
1 2 (y
− 1)t
´s
∞ X (1 + α1 + β1 )r
−1−α1 −β1
= e (1 − u)
m!
m=0
r!(1 + α1 )r
r=0
Ã
1 2 (x − 1)t
(1 − u)
!r
³ ´s ∞ ∞ P P (1+α2 +β2 )k+s (λ)k k 1 v (y − 1)t
k=0 s=0
t
−1−α1 −β1
= e (1 − u) ×Φ1
2
k!s!(1+α2 )k+s
∙
1 F1
"
1 2 (x
− 1)t 1 + α1 + β1 ; +α1 ; (1 − u)
#
¸
1 1 + α2 + β2 ; λ; 1 + α2 ; v, (y − 1)t 2
which proves (2.1). The proof of results (2.2) to (2.8) are similar to that of (2.1). PROOF OF (2.9) ∞ ∞ X ∞ X X n!(λ)m (1 + α1 + β1 )m (µ)k (1 + α2 + β2 )k
m!k!(1 + α1 )m+n (1 + α2 )k+n
m=0 k=0 n=0
×Pn(α1 +m, β1 −n; α2 +k, β2 −n) (x, y)tn um vk =
∞ ∞ X ∞ X X
n!(λ)m (1 + α1 + β1 )m (µ)k (1 + α2 + β2 )k m!k!(1 + α1 + m)n (1 + α1 )m (1 + α2 + k)n (1 + α2 )k m=0 k=0 n=0 ×
(1 + α1 + m)n (1 + α2 + k)n (n!)2
228
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
× × =
³
1−x 2
n n−r X X (−n)r+s (1 + α1 + β1 + m)r (1 + α2 + β2 + k)s
r!s!(1 + α1 + m)r (1 + α2 + k)s
r=0 s=0
´r ³
´ 1−y s n m k t u v 2
∞ X ∞ X ∞ X n n−r X X (−1)r+s n!(λ)m (1 + α1 + β1 )m+r (µ)k (1 + α2 + β2 )k+s
m!k!n!r!s!(n − r − s)!(1 + α1 )m+r (1 + α2 )k+s
m=0 k=0 n=0 r=0 s=0
× =
µ
1−x 2
¶r µ
1−y 2
¶s
× tn um v k
∞ X ∞ X ∞ X ∞ ∞ X X (λ)m (1 + α1 + β1 )m+r (µ)k (1 + α2 + β2 )k+s
m!k!n!r!s!(1 + α1 )m+r (1 + α2 )k+s
m=0 k=0 n=0 r=0 s=0
µ
x−1 × 2 = et
¶r µ
y−1 2
¶s
tn+r+s um vk
µ ¶r ∞ X ∞ X (1 + α1 + β1 )m+r (λ)m m 1 (x − 1)t u
m!r!(1 + α1 )m+r
m=0 r=0
2
µ ¶s ∞ ∞ X X (1 + α2 + β2 )k+s (µ)k k 1 (y − 1)t v
k!s!(1 + α2 )k+s
k=0 s=0
∙
1 1 + α1 + β1 , λ; 1 + α1 ; u, (x − 1)t 2
t
= e Φ1 ×Φ1
2
∙
1 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t 2
¸
¸
which proves (2.9). The proof of results (2.10) to (2.40) are similar to that of (2.9). Further we have the following triple generating functions of Jacobi polynomials of two variables: ∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 )n (1 + α2 )n
Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k
Triple generating functions of Jacobi polynomials of two variables 229 "
= et (1−u)−1−α1 −β1 (1−v)−1−α2 −β2 1 F1 1 + α1 + β1 ; 1 + α1 ;
(2.41)
×1 F1
"
1 2 (y
m!k!(1 + α1 )n (1 + α2 + k)n
t
−1−α1 −β1
= e (1 − u)
#
1 F1
Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um v k
"
1 + α1 + β1 ; 1 + α1 ;
1 2 (x − 1)t
(1 − u)
∙
#
¸
1 × Φ1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
(2.42)
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m!k!(1 + α1 + m)n (1 + α2 )n
m=0 k=0 n=0 t
(1 − u)
#
− 1)t 1 + α2 + β2 ; 1 + α2 ; , (1 − v)
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
1 2 (x − 1)t
−1−α2 −β2
= e (1−v)
(2.43)
×Φ1
1 F1
"
Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um v k 1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
# ¸
1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, (x − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α + k) 1 n 2 n m=0 k=0 n=0 t
−1−α1 −β1
= e (1 − u)
1 F1
"
1 + α1 + β1 ; 1 + α1 ;
∙
1 2 (x − 1)t
(1 − u)
#
¸
1 Φ2 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
(2.44) ∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um vk m!k!(1 + α + m) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
−1−α2 −β2
= e (1 − v)
1 F1
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
#
230
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
(2.45)
× Φ2
∙
¸
1 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 = et (1 − u)−1−α1 −β1 1 F1 (2.46)
× Φ3
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
∙
#
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 n m=0 k=0 n=0 = et (1 − v)−1−α2 −β2 1 F1 (2.47)
× Φ3
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u , 2
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (λ)k
m!k!(1 + α1 )n (1 + α2 )n m=0 k=0 n=0 "
Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k
= et (1−u)−1−α1 −β1 1 F1 1 + α1 + β1 ; 1 + α1 ;
(2.48)
×Ψ1
−1−α2 −β2
= e (1−v)
×Ψ1
(1 − u)
# ¸
1 1 + α2 + β2 , λ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
m!k!(1 + α1 )n (1 + α2 )n m=0 k=0 n=0
(2.49)
1 2 (x − 1)t
∙
∞ ∞ X ∞ X X n!(λ)m (1 + α2 + β2 )k t
#
∙
1 F1
"
Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k 1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
# ¸
1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 231
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
−1−α1 −β1
= e (1−u)
(2.50)
×Ψ2
1 F1
"
1 + α1 + β1 ; 1 + α1 ;
1 2 (x − 1)t
(1 − u)
#
∙
¸
1 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 = et (1−v)−1−α2 −β2 1 F1
(2.51)
×Ψ2
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
∙
¸
1 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t , 2
∞ X ∞ ∞ X X n!(1 + α1 + β1 )m (λ)k (µ)k
m!k!(1 + α1 )n (1 + α2 )k+n m=0 k=0 n=0 =et
(2.52)
(1 − u)−1−α1 −β1
1 F1
Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk
∙
1 + α1 + β1 ; 1 + α1 ;
(1−u)
∙
¸
¸
m!k!(1 + α1 )m+n (1 + α2 )n m=0 k=0 n=0
(2.53)
1 (x−1)t 2
1 × Ξ1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X n!(λ)m (µ)m (1 + α2 + β2 )k
=et
#
(1 − v)−1−α2 −β2 Ξ1
∙
1 F1
∙
Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k
1 + α2 + β2 ; 1 + α2 ;
1 (y−1)t 2
(1−v)
¸
¸
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t , 2
232
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
m!k!(1 + α1 + m)n (1 + α2 + k)n
Pn(α1 +m, β1 −n; α2 +k, β2 −n) (x, y)tn um v k
h
=et Φ1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, 12 (x − 1)t (2.54)
× Φ1
∙
i
¸
1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n! m!k!(1 + α + m) 1 n (1 + α2 + k)n m=0 k=0 n=0 h
×Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k
=et Φ2 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, 12 (x − 1)t ∙
i
¸
1 × Φ2 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
(2.55)
∞ ∞ X ∞ X X
m=0 k=0 n=0
n! m!k!(1 + α1 )m+n (1 + α2 )k+n
(α1 +m, β1 −m−n; α2 +k, β2 −k−n)
Pn
t
= e Φ3
(2.56)
Φ3
∞ ∞ X ∞ X X
∙
(x, y)tn um vk
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2 ¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
n!(λ)m (µ)k Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!((1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
= e Ψ1
∙
¸
1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2
Triple generating functions of Jacobi polynomials of two variables 233 h
i
×Ψ1 1 + α2 + β2 , µ; 1 + α2 + β2 , 1 + α2 ; v, 12 (y − 1)t , (2.57) ∞ ∞ X ∞ X X
n! Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 ∙
1 = et Ψ2 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 (2.58)
×Ψ2
∙
¸ ¸
1 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
∞ X ∞ ∞ X X
n!(λ)m (µ)m (δ)k (ν)k Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 k+n m=0 k=0 n=0 ∙
1 = et Ξ1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2 (2.59)
∙
¸
¸
1 × Ξ1 δ, 1 + α2 + β2 , ν; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α + m) (1 + α + k) 1 n 2 n m=0 k=0 n=0 t
= e Φ1
(2.60)
∙
1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 × Φ2 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k Pn(α1 +m, β1 −m−n; α2 +k, β2 −n) (x, y)tn um v k m!k!(1 + α + m) (1 + α + k) 1 n 2 n m=0 k=0 n=0 t
= e Φ2
(2.61)
× Φ1
∙
∙
1 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2 ¸
¸
1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
234
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α + m) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 t
= e Φ1
(2.62)
∙
1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, (x − 1)t 2 ∙
¸
¸
1 × Φ3 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2 ∞ ∞ X ∞ X X
n!(1 + α2 + β2 )k m!k!(1 + α1 )m+n (1 + α2 + k)n m=0 k=0 n=0 (α1 +m, β1 −m−n; α2 +k, β2 −n)
Pn
t
= e Φ3
(2.63)
× Φ1
(x, y)tn um v k
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
∙
¸
1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n! m!k!(1 + α1 + m)n (1 + α2 )k+n m=0 k=0 n=0 (α1 +m, β1 −m−n; α2 +k, β2 −k−n)
Pn
(x, y)tn um vk
∙
¸
1 = et Φ2 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2 (2.64)
×Φ3
∙
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
∞ ∞ X ∞ X X
m=0 k=0 n=0
n! m!k!(1 + α1 )m+n (1 + α2 + k)n
(α1 +m, β1 −m−n; α2 +k, β2 −k−n)
Pn
(x, y)tn um vk
Triple generating functions of Jacobi polynomials of two variables 235
t
= e Φ3
(2.65)
× Φ2
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
∙
¸
1 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m (λ)k Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α + m) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
= e Φ1
∙
¸
1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, (x − 1)t 2
(2.66)
×Ψ1
∙
¸
1 1 + α2 + β2 , λ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)m (1 + α2 + β2 )k Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk m!k!(1 + α ) (1 + α + k) 1 n 2 n m=0 k=0 n=0 ∙
¸
1 = et Ψ1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 ∙
¸
1 ×Φ1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
(2.67)
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m Pn(α1 +m, β1 −n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α + m) (1 + α ) 1 n 2 n m=0 k=0 n=0 ∙
1 = et Φ1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, (x − 1)t 2 (2.68)
∞ ∞ X ∞ X X
∙
¸ ¸
1 ×Ψ2 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
n!(1 + α2 + β2 )k Pn(α1 , β1 +m−n; α2 +k, β2 −n) (x, y)tn um vk m!k!(1 + α ) (1 + α + k) 1 n 2 n m=0 k=0 n=0
236
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas t
= e Ψ2
(2.69)
∙
1 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2
×Φ1
∙
¸
¸
1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(1 + α1 + β1 )m (λ)k (µ)k m!k!(1 + α1 + m)n (1 + α2 )k+n m=0 k=0 n=0 Pn(α1 +m, β1 −n; α2 +k, β2 −k−n) (x, y)tn um v k
h
=et Φ1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, 12 (x − 1)t
(2.70)
i
∙
¸
1 × Ξ1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2 ∞ ∞ X ∞ X X
n!(λ)m (µ)m (1 + α2 + β2 )k m!k!(1 + α1 )m+n (1 + α2 + k)n m=0 k=0 n=0 Pn(α1 +m, β1 −m−n; α2 +k, β2 −n) (x, y)tn um v k ∙
1 = et Ξ1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2 (2.71)
∙
¸
¸
1 ×Φ1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)k Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um vk m!k!(1 + α + m) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
= e Φ2
(2.72)
×Ψ1
∙
∙
1 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
¸ ¸
1 1 + α2 + β2 , λ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 237
∞ X ∞ ∞ X X
n!(λ)m Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α + k) 1 n 2 n m=0 k=0 n=0 t
= e Ψ1
∙
¸
1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 ∙
¸
1 ×Φ2 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
(2.73)
∞ ∞ X ∞ X X
n! Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um vk m!k!(1 + α + m) (1 + α ) 1 n 2 n m=0 k=0 n=0 ∙
1 = et Φ2 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
¸
∙
¸
1 ×Ψ2 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
(2.74)
∞ ∞ X ∞ X X
n! Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α + k) 1 n 2 n m=0 k=0 n=0 ∙
1 = et Ψ2 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 (2.75)
×Φ2
∙
¸
¸
1 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
∞ ∞ X ∞ X X
n!(λ)k (µ)k Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α + m) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 ∙
1 1 + α1 , 1 + α1 + β1 ; 1 + α1 ; u, (x − 1)t 2
t
= e Φ2
(2.76)
Ξ1
∙
¸
1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
¸
238
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
∞ X ∞ ∞ X X
n!(λ)m (µ)m Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α + k) 1 m+n 2 n m=0 k=0 n=0 t
= e Ξ1
∙
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
∙
¸
¸
1 Φ2 1 + α2 , 1 + α2 + β2 ; 1 + α2 ; v, (y − 1)t , 2
(2.77)
∞ ∞ X ∞ X X
n!(λ)k Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 n m=0 k=0 n=0 ∙
¸
1 = et Φ3 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2 ∙
¸
1 Ψ1 1 + α2 + β2 , λ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
(2.78)
∞ ∞ X ∞ X X
n!(λ)m Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 ∙
¸
1 = et Ψ1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 (2.79)
Φ3
∙
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
∞ ∞ X ∞ X X
n! Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 n m=0 k=0 n=0 t
= e Φ3
(2.80)
× Ψ2
∙
∙
¸
1 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2
¸
1 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
Triple generating functions of Jacobi polynomials of two variables 239
∞ X ∞ ∞ X X
n! Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 t
= e Ψ2
∙
1 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 ∙
¸
¸
1 × Φ3 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
(2.81)
∞ ∞ X ∞ X X
n!(λ)k (µ)k Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 k+n m=0 k=0 n=0 ∙
¸
1 = et Φ3 1 + α1 + β1 ; 1 + α1 ; (x − 1)t, u 2 ∙
¸
1 Ξ1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
(2.82)
∞ ∞ X ∞ X X
n!(λ)m (µ)m Pn(α1 +m, β1 −m−n; α2 +k, β2 −k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 k+n m=0 k=0 n=0 ∙
1 = et Ξ1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2 (2.83)
Φ3
∞ ∞ X ∞ X X
∙
¸
¸
1 1 + α2 + β2 ; 1 + α2 ; (y − 1)t, v , 2
n!(λ)m Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!((1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
= e Ψ1
(2.84)
∙
¸
1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 ×Ψ2
∙
¸
1 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
240
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
∞ X ∞ ∞ X X
n!(λ)k Pn(α1 , β1 +m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 n 2 n m=0 k=0 n=0 t
= e Ψ2
∙
1 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2
¸
∙
¸
1 ×Ψ1 1 + α2 + β2 , λ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
(2.85)
∞ ∞ X ∞ X X
n!(λ)m (µ)k (δ)k Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 ∙
¸
1 = et Ψ1 1 + α1 + β1 , λ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2 ∙
¸
1 ×Ξ1 µ, 1 + α2 + β2 , δ; 1 + α2 ; v, (y − 1)t , 2
(2.86)
∞ ∞ X ∞ X X
n!(λ)m (µ)m (δ)k Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 n m=0 k=0 n=0 ∙
1 = et Ξ1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2 (2.87)
×Ψ1
∞ ∞ X ∞ X X
¸
∙
¸
1 1 + α2 + β2 , δ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t , 2
n!(λ)k (µ)k Pn(α1 , β1 +m−n; α2 +k, β2 −k−n) (x, y)tn um vk m!k!(1 + α ) (1 + α ) 1 n 2 k+n m=0 k=0 n=0 t
= e Ψ2
(2.88)
∙
1 1 + α1 + β1 ; 1 + α1 + β1 , 1 + α1 ; u, (x − 1)t 2
× Ξ1
∙
¸
1 λ, 1 + α2 + β2 , µ; 1 + α2 ; v, (y − 1)t , 2
¸
Triple generating functions of Jacobi polynomials of two variables 241
∞ ∞ X ∞ X X
n!(λ)m (µ)m Pn(α1 +m, β1 −m−n; α2 , β2 +k−n) (x, y)tn um v k m!k!(1 + α ) (1 + α ) 1 m+n 2 n m=0 k=0 n=0 t
= e Ξ1
(2.89)
×Ψ2
∙
1 λ, 1 + α1 + β1 , µ; 1 + α1 ; u, (x − 1)t 2
¸
∙
¸
1 1 + α2 + β2 ; 1 + α2 + β2 , 1 + α2 ; v, (y − 1)t . 2
PROOF OF (2.41) ∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m!k!(1 + α1 )n (1 + α2 )n
m=0 k=0 n=0
=
Pn(α1 ,β1 +m−n;α2 ,β2 +k−n) (x, y)tn um vk
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
×
m!k!(1 + α1 )n (1 + α2 )n
r!s!(1 + α1 )r (1 + α2 )s
r=0 s=0
1−x × 2
× =
1−y 2
¶s
tn um v k
µ
m!k!n!r!s!(n − r − s)!(1 + α1 )r (1 + α2 )s
1−x 2
¶r µ
1−y 2
¶s
tn um v k
∞ X ∞ X ∞ X ∞ X ∞ X (1 + α1 + β1 )m+r (1 + α2 + β2 )k+s
m=0 k=0 n=0 r=0 s=0
µ
x−1 × 2 =e
¶r µ
∞ X ∞ X ∞ X n n−r X X (−1)r+s n!(1 + α1 + β1 )m+r (1 + α2 + β2 )k+s
m=0 k=0 n=0 r=0 s=0
t
(1 + α1 )n (1 + α2 )n (n!)2
n n−r X X (−n)r+s (1 + α1 + β1 + m)r (1 + α2 + β2 + k)s
µ
=
×
m!k!n!r!s!(1 + α1 )r (1 + α2 )s
¶r µ
y−1 2
¶s
tn+r+s um vk
µ ∞ X ∞ X (1 + α1 + β1 )r (1 + α2 + β2 )s 1
r=0 s=0
r!s!(1 + α1 )r (1 + α2 )s
2
(x − 1)t
¶r µ
1 (y − 1)t 2
¶s
242
Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
×
∞ ∞ X (1 + α1 + β1 + r)m m X (1 + α2 + β2 + s)k k u v
m!
m=0
t
−1−α1 −β1
= e (1 − u)
×
t
−1−α1 −β1
= e (1 − u)
∞ X (1 + α1 + β1 )r
−1−α2 −β2
(1 − v)
s!(1 + α2 )s
−1−α2 −β2
(1 − v)
× 1 F1
r!(1 + α1 )r
r=0
∞ X (1 + α2 + β2 )s
s=0
k!
k=0
Ã
1 F1
1 2 (y
− 1)t (1 − v)
Ã
1 2 (x − 1)t
(1 − u)
!r
!s
"
1 2 (x
− 1)t 1 + α1 + β1 ; 1 + α1 ; (1 − u)
"
1 2 (y
− 1)t 1 + α2 + β2 ; 1 + α2 ; (1 − v)
#
#
which proves (2.41). The proof of results (2.42) to (2.53) are similar to that of (2.41). PROOF OF (2.54) : ∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m!k!(1 + α1 + m)n (1 + α2 + k)n m=0 k=0 n=0
=
∞ ∞ X ∞ X X n!(1 + α1 + β1 )m (1 + α2 + β2 )k
m=0 k=0 n=0
× × =
Pn(α1 +m,β1 −n;α2 +k,β2 −n) (x, y)tn um v k
³
1−x 2
´r ³
m!k!(1 + α1 + m)n (1 + α2 + k)n
×
(1 + α1 + m)n (1 + α2 + k)n (n!)2
n n−r X X (−n)r+s (1 + α1 + β1 + m)r (1 + α2 + β2 + k)s
r=0 s=0
1−y 2
´s
r!s!(1 + α1 + m)r (1 + α2 + k)s
tn um vk
∞ X ∞ X ∞ X n n−r X X (−1)r+s n!(1 + α1 + β1 )m+r (1 + α2 + β2 )k+s (1 + α1 )m (1 + α2 )k
m=0 k=0 n=0 r=0 s=0
m!k!n!r!s!(n − r − s)!(1 + α1 )m+r (1 + α2 )k+s µ
1−x 2
¶r
Triple generating functions of Jacobi polynomials of two variables 243 µ
1−y × 2 =
¶s
tn um vk
∞ X ∞ X ∞ X ∞ X ∞ X (1 + α1 + β1 )m+r (1 + α2 + β2 )k+s (1 + α1 )m (1 + α2 )k
m=0 k=0 n=0 r=0 s=0
³
x−1 2
=et
´r ³
´ y−1 s 2
m!k!n!r!s!(1 + α1 )m+r (1 + α2 )k+s
× tn+r+s um v k
³ ´r ∞ P ∞ P (1+α1 +β1 )m+r (1+α1 )m m 1 u (x − 1)t
m!r!(1+α1 )m+r m=0 r=0 ∞ ∞ P P (1+α2 +β2 )k+s (1+α2 )k × k!s!(1+α2 )k+s k=0 s=0
h
2
vk
³
1 2 (y
− 1)t
´s
=et Φ1 1 + α1 + β1 , 1 + α1 ; 1 + α1 ; u, 12 (x − 1)t h
Φ1 1 + α2 + β2 , 1 + α2 ; 1 + α2 ; v, 12 (y − 1)t
i
i
which proves (2.54). The proof of results (2.55) to (2.89) are similar to that of (2.54).
References [1] Chatterjea, S. K., A note on Laguerre polynomials of two variables, Bull. cal. Math. Soc. 83, (1991). [2] Khan, M. A. and Abukhammash, G. S., On Hermite polynomials of two variables suggested by S. F. Ragab’s Laguerre polynomials of two variables, Bull. cal. Math. Soc. 90, pp. 61-76, (1998). [3] Khan, M. A. and Shukla, A. K., On Laguerre polynomials of several variables,Bull. cal. Math. Soc. 89, (1997). [4] Khan, M. A., A note on Laguerre polynomials of m- variables,Bull.of the Greek. Math.Soc. 40, pp. 113-117, (1998). [5] Khan,M. A.,Khan, A. H. and Abbas,S.M.,A note on Pseudo two variables Jacobi polynomials ,International transactions in applied sciences, (2010)
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Mumtaz A. Khan, Abdul H. Khan Sayed M. Abbas
[6] Khan, M. A., Khan, A. H. and Abbas, S. M.,On some generating and double generating functions of Jacobi polynomials of two variables, Communicated for publication. (α,β)
[7] Ragab, S. F. On Laguerre polynomials of two Ln ables,Bull. cal. Math. Soc. 83, (1991).
(x, y) vari-
[8] Rainville, E. D. Special Functions, Chelsea Publishing Company, Bronx, New York, (1971). [9] Srivastava, H. M. and Manocha, H. L., A treatise on generating functions,John Wily and Sons (Halsted Press, New York, Ellis Horwood, Chichester). [10] Shrivastava, H. S. P., On Jacobi polynomials of several variables, Integral Transforms and Special functions 10 No. 1, pp. 61-70, (2000). [11] Shrivastava, H. S. P., Generating function,Expansion and Recurrence Relations of Jacobi polynomials of two variables, Acta Ciencia No. 4, pp. 633-640, (2002). [12] Shrivastava, H. S. P.,Derivatives of multivariable Jacobi polynomials,Acta Ciencia No. 3, pp. 451-458, (2002).
Mumtaz Ahmad Khan Department of Applied Mathematics Faculty of Engineering and Technology Aligarh Muslim University Aligarh - 202002, U.P. India e-mail : mumtaz ahmad khan
[email protected] Abdul Hakim Khan Department of Applied Mathematics Faculty of Engineering and Technology Aligarh Muslim University Aligarh - 202002, U.P. India e-mail :
[email protected] and
Triple generating functions of Jacobi polynomials of two variables 245 Sayed Mohammad Abbas Department of Applied Mathematics Faculty of Engineering and Technology Aligarh Muslim University Aligarh - 202002, U.P. India e-mail : syedmohammad
[email protected] 
