88 ON q-LAPLACE TRANSFORMS OF A GENERAL CLASS OF q ...

84 R. K. YADAV, S. D. PUROHIT AND P. NIRWAN
polynomials, the q-Bessel polynomials of second type, the q-Laguerre polynomials,
the q-Jacobi polynomials, the q-Konhauser polynomials and several others. We
mention the definitions of some of these polynomials as under
The q-Rogers- Szegö polynomials
n∑
[ ] n
h n (x ; q) = x k . (1.23)
k
q
The discrete q-Hermite polynomials
H n (x ; q) =
[n/2]
∑
k=0
k=0
The Generalized Stieltjes-Weigert polynomials
∑
n [ ]
S n (x, p; q) = (−1) n q −n(2n+1)/2 n
(p; q) n
j
The q-Bessel polynomials of second type
(q; q) n (−1) k q k(k−1) x n−2k
(q 2 ; q 2 ) k (q; q) n−2k
· (1.24)
j=0
y n (x; α/q 2 ) = q n(n−1)/2 2Φ 1
[ q −n , q α+n−1 ;
−q ;
q
· qj2 (−q 1/2 x) j
(p; q) j
· (1.25)
]
q , −2xq . (1.26)
We shall also use the following definitions of various q-Polynomials and basic hypergeometric
functions cf. Gasper and Rahman [7], Jain and Srivastava [9], Koelink
and Swarttouw [10], in the sequel.
The Affine q-Krowtchauck polynomials
K Aff
n (x; a, N; q) = 3 Φ 2
[ q −n , x, 0;
aq , q −N ;
The Haln-Exton q-Bessel function J v (x ; q)
J v (x ; q) = xv (q v+1 [
; q) ∞ 0;
· 1Φ 1
(q : q) ∞ q v+1 ;
The big q-Jacobi polynomials
P n
(α,β) (x; γ, δ; q) = (αq, −δαq/γ; q) n(γ/αq) n
·
(q, −q; q) n
[
q
3Φ −n , α β q n+1 , α xq/γ ;
2
αq, −δα q/γ ;
The q-Lommel polynomials
m∑
]
q, q . (1.27)
q , qx 2 ]
. (1.28)
]
q, q . (1.29)
x 2n−m (q n+1 ; q) ∞ (q v [
; q) ∞ q
R m,v (x ; q) =
(q; q)
n=0 ∞ (q v+m−n · −n , q v+m−n ;
2Φ 1
; q) ∞ q v q , q
].
n+1
;
(1.30)
The q-Bessel function of second type is given by
J −v (x; q) = eivπ (q v+1 ; q) ∞ x −v q v(v−1)/2 ∞∑ (−1) k q k(k+1)/2 x 2k q −vk
·
(q; q) ∞ (q −v+1 . (1.31)
, q) k (q, q) k
k=0