88 ON q-LAPLACE TRANSFORMS OF A GENERAL CLASS OF q ...
88 ON q-LAPLACE TRANSFORMS OF A GENERAL CLASS OF q ...
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Math. Maced.<br />
Vol. 7 (2009)<br />
81– <strong>88</strong><br />
<strong>ON</strong> q-<strong>LAPLACE</strong> <strong>TRANSFORMS</strong> <strong>OF</strong> A <strong>GENERAL</strong> <strong>CLASS</strong><br />
<strong>OF</strong> q-POLYNOMIALS AND q-HYPERGEOMETRIC FUNCTI<strong>ON</strong>S<br />
R. K. YADAV ∗ , S. D. PUROHIT ∗∗ AND PO<strong>ON</strong>AM NIRWAN ∗<br />
Abstract. This paper invisage the derivation of a theorem concerning the q-<br />
Laplace image of a class of q-polynomial family and its applications in terms<br />
of the q-Laplace transforms of various q-polynomials. Results involving the<br />
q-Laplace transforms of q-polynomials in terms of the basic Kampé-de-Fériet<br />
functions are also deduced.<br />
1. Introduction and Definitions<br />
Recently, Yadav and Purohit [15]-[17] have evaluated the q-Laplace images of<br />
number of q-polynomials and basic hypergeometric functions of one and more<br />
variables with several interesting special cases. This has pawed a way to further<br />
investigate the q-Laplace transform of general class of q-polynomial and basic<br />
hypergeometric functions.<br />
Hahn [8], defined the q-analogues of the well known classical Laplace transform:<br />
ϕ(s) =<br />
∫ ∞<br />
by means of the following two q-integrals:<br />
and<br />
qL s {f(t)} =<br />
qL s {f(t)} =<br />
1<br />
(1 − q)<br />
0<br />
1<br />
(1 − q)<br />
∫ ∞<br />
0<br />
e −st f(t) dt , (1.1)<br />
∫ s−1<br />
0<br />
E q (qst)f(t) d(t; q), (1.2)<br />
e q (−st)f(t) d(t; q); Re(s) > 0. (1.3)<br />
Where the q-exponential series is defined as:<br />
∞∑ x n<br />
e q (x) =<br />
. (1.4)<br />
(q ; q) n<br />
n=0<br />
2000 Mathematics Subject Classification. Primary:33D60, Secondary: 44A10, 44A20.<br />
Key words and phrases. q-Laplace transforms, general class of q-polynomials, basic hypergeometric<br />
functions, and basic Kampé-de Fériet functions.<br />
81
82 R. K. YADAV, S. D. PUROHIT AND P. NIRWAN<br />
And<br />
E q (x) =<br />
∞∑ (−1) n q n(n−1)/2 x n<br />
, (1.5)<br />
(q ; q) n<br />
n=0<br />
The basic integrals cf. Gasper and Rahman [7], are defined as:<br />
∫ x<br />
0<br />
∫ ∞<br />
x<br />
∫ ∞<br />
0<br />
f(t) d(t ; q) = x(1 − q)<br />
f(t) d(t ; q) = x(1 − q)<br />
f(t) d(t ; q) = (1 − q)<br />
∞∑<br />
q k f(xq k ), (1.6)<br />
k=0<br />
∞∑<br />
q −k f(xq −k ), (1.7)<br />
k=1<br />
∞∑<br />
k=−∞<br />
q k f(q k ). (1.8)<br />
By virtue of the results (1.6), the integral equation (1.2) can be expressed as:<br />
ϕ(s) ≡ q L s {f(t)} = (q; q) ∞<br />
s<br />
∞∑<br />
j=0<br />
q j f(s −1 q j )<br />
(q; q) j<br />
. (1.9)<br />
Where the function ϕ(s) is called as q-Laplace transform or q-image of the original<br />
functions f(t).<br />
For real, or complex α and 0 < |q| < 1 , the q-shifted factorial is defined by:<br />
(α ; q) n<br />
=<br />
{<br />
1, if n = 0<br />
(1 − α)(1 − αq) · · · (1 − αq n−1 ) , if n ∈ N.<br />
(1.10)<br />
Also, for x ≠ 0, we have<br />
and<br />
[x − y] ν<br />
= x ν ∞ ∏<br />
n=o<br />
where α ≠ 0, −1, −2, · · · .<br />
The q-binomial series is given by<br />
{ (1 −<br />
y }<br />
x qn )<br />
(1 − y = x ν ( y<br />
x qν+n )<br />
x ; q) ν, (1.11)<br />
Γ q (α) = (q; q) ∞(1 − q) 1−α<br />
(q α ; q) ∞<br />
, (1.12)<br />
1Φ 0 (α; − ; q, x) = (αx; q) ∞<br />
(x; q) ∞<br />
. (1.13)<br />
The q-binomial coefficients are given by<br />
[ ]<br />
n (q; q) n<br />
=<br />
. (1.14)<br />
k (q; q) n−k (q; q) k<br />
q
<strong>ON</strong> q-<strong>LAPLACE</strong> <strong>TRANSFORMS</strong> <strong>OF</strong> A <strong>GENERAL</strong> <strong>CLASS</strong>... 83<br />
The generalized basic hypergeometric function cf. Gasper and Rahman [7], is given<br />
by<br />
[ ]<br />
a1 , · · · , a r ;<br />
∞∑ (a 1 , · · · , a r ; q) n x n {<br />
}<br />
rΦ s q, x =<br />
(−1) n q n(n−1) (1+s−r)<br />
2<br />
,<br />
b 1 , · · · , b s ;<br />
(q, b<br />
n=0 1 , · · · , b s ; q) n<br />
(1.15)<br />
where, for convergence, we have 0 < |q| < 1 , |x| < 1 if r = s + 1, and for any x if<br />
r ≤ s.<br />
Another form of the generalized basic hypergeometric series r Φ s (.) cf. Slater [14],<br />
is defined as<br />
rΦ s<br />
[<br />
a1 , · · · , a r ;<br />
b 1 , · · · , b s ;<br />
]<br />
q, x =<br />
∞∑<br />
n=0<br />
(a 1 , · · · , a r ; q) n x n<br />
(b 1 , · · · , b s ; q) n (q; q) n<br />
. (1.16)<br />
The basic analogue of Kampé-de Fériet function cf. Srivastava and Karlsson [13],<br />
is defined as<br />
( )<br />
A: B; B′ (a) : (b); (b<br />
Φ ′ );<br />
C: D;D ′ (c) : (d); (d ′ ); q; x, y<br />
= ∑<br />
r,s≥0<br />
A∏<br />
j=1<br />
∏B ′<br />
(a j ; q) r+s<br />
j=1<br />
B ∏<br />
′′<br />
(b j ; q) r<br />
j=1<br />
(b ′ j ; q) s x r y s<br />
, (1.17)<br />
C∏<br />
D ∏<br />
′<br />
D ∏<br />
′′<br />
(c j ; q) r+s (d j ; q) r (d ′ j ; q) s (q; q) r (q; q) s<br />
j=1<br />
j=1<br />
j=1<br />
where, for convergence |x| < 1, |y| < 1 and 0 < |q| < 1.<br />
The basic sine and cosine series are defined as<br />
sin q (x) = e q(ix) − e q (−ix)<br />
,<br />
2i<br />
(1.18)<br />
cos q (x) = e q(ix) + e q (−ix)<br />
.<br />
2<br />
(1.19)<br />
The basic sine hyperbolic and basic cosine hyperbolic series are defined as<br />
sinh q (x) = e q(x) − e q (−x)<br />
, (1.20)<br />
2<br />
cosh q (x) = e q(x) + e q (−x)<br />
. (1.21)<br />
2<br />
The general classes of family of basic hypergeometric polynomials f n,N (x; q) cf.<br />
Srivastava and Agarwal [12], in terms of a sequence {S j,q } N j=0<br />
of parameters is<br />
defined as<br />
f n,N (x; q) =<br />
[n/N]<br />
∑<br />
j=0<br />
[<br />
n<br />
Nj<br />
]<br />
S j,q x j , (1.22)<br />
where N is positive integer and n = 0, 1, 2, · · · .<br />
On suitable specialization of the sequence of arbitrary parameters S j,q , the q-<br />
Polynomial family f n,N (x; q) yields a number of known q-Polynomials cf. Gasper<br />
and Rahman [7], as its special cases. These include namely, the q-Rogers-Szegö<br />
polynomials, the discrete q-Hermite polynomials, the Generalized Stieltjes-Weigert
84 R. K. YADAV, S. D. PUROHIT AND P. NIRWAN<br />
polynomials, the q-Bessel polynomials of second type, the q-Laguerre polynomials,<br />
the q-Jacobi polynomials, the q-Konhauser polynomials and several others. We<br />
mention the definitions of some of these polynomials as under<br />
The q-Rogers- Szegö polynomials<br />
n∑<br />
[ ] n<br />
h n (x ; q) = x k . (1.23)<br />
k<br />
q<br />
The discrete q-Hermite polynomials<br />
H n (x ; q) =<br />
[n/2]<br />
∑<br />
k=0<br />
k=0<br />
The Generalized Stieltjes-Weigert polynomials<br />
∑<br />
n [ ]<br />
S n (x, p; q) = (−1) n q −n(2n+1)/2 n<br />
(p; q) n<br />
j<br />
The q-Bessel polynomials of second type<br />
(q; q) n (−1) k q k(k−1) x n−2k<br />
(q 2 ; q 2 ) k (q; q) n−2k<br />
· (1.24)<br />
j=0<br />
y n (x; α/q 2 ) = q n(n−1)/2 2Φ 1<br />
[ q −n , q α+n−1 ;<br />
−q ;<br />
q<br />
· qj2 (−q 1/2 x) j<br />
(p; q) j<br />
· (1.25)<br />
]<br />
q , −2xq . (1.26)<br />
We shall also use the following definitions of various q-Polynomials and basic hypergeometric<br />
functions cf. Gasper and Rahman [7], Jain and Srivastava [9], Koelink<br />
and Swarttouw [10], in the sequel.<br />
The Affine q-Krowtchauck polynomials<br />
K Aff<br />
n (x; a, N; q) = 3 Φ 2<br />
[ q −n , x, 0;<br />
aq , q −N ;<br />
The Haln-Exton q-Bessel function J v (x ; q)<br />
J v (x ; q) = xv (q v+1 [<br />
; q) ∞ 0;<br />
· 1Φ 1<br />
(q : q) ∞ q v+1 ;<br />
The big q-Jacobi polynomials<br />
P n<br />
(α,β) (x; γ, δ; q) = (αq, −δαq/γ; q) n(γ/αq) n<br />
·<br />
(q, −q; q) n<br />
[<br />
q<br />
3Φ −n , α β q n+1 , α xq/γ ;<br />
2<br />
αq, −δα q/γ ;<br />
The q-Lommel polynomials<br />
m∑<br />
]<br />
q, q . (1.27)<br />
q , qx 2 ]<br />
. (1.28)<br />
]<br />
q, q . (1.29)<br />
x 2n−m (q n+1 ; q) ∞ (q v [<br />
; q) ∞ q<br />
R m,v (x ; q) =<br />
(q; q)<br />
n=0 ∞ (q v+m−n · −n , q v+m−n ;<br />
2Φ 1<br />
; q) ∞ q v q , q<br />
].<br />
n+1<br />
;<br />
(1.30)<br />
The q-Bessel function of second type is given by<br />
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<br />
·<br />
(q; q) ∞ (q −v+1 . (1.31)<br />
, q) k (q, q) k<br />
k=0
<strong>ON</strong> q-<strong>LAPLACE</strong> <strong>TRANSFORMS</strong> <strong>OF</strong> A <strong>GENERAL</strong> <strong>CLASS</strong>... 85<br />
2. Main Result<br />
This section envisage to derive a theorem involving the q-Laplace transforms<br />
of the general class of q-Polynomials and certain basic hypergeometric functions.<br />
Interestingly, some of the results are obtained in terms of the q-analogue of the<br />
Kampé-de-Fériet functions.<br />
Theorem 1. Let f n,N (x k ; q) be the family of q-Polynomials defined in terms of<br />
a sequence S j,q (.) of complex coefficients, then the following result involving the<br />
q-Laplace transform of the x λ -weighted family of q-Polynomials holds:<br />
{<br />
qL s x λ f n,N (x k ; q) } [n/N]<br />
(1 − q)λ ∑<br />
[ ] ( ) kj<br />
n 1 − q<br />
=<br />
s λ+1<br />
S<br />
Nj j,q Γ q (kj + λ + 1),<br />
j=0<br />
q<br />
s<br />
(2.1)<br />
where Re(k + λ + 1) > 0, λ > 0, k ∈ I, and N is a positive integer.<br />
Proof. We employ (1.9) and (1.22) in the left hand side of the result (2.1), to<br />
obtain<br />
{<br />
qL s x λ f n,N (x k ; q) } = (q; q) ∑ ∞<br />
∞ q i(1+λ) [n/N]<br />
∑<br />
[ ] ( )<br />
n q<br />
i kj<br />
s λ+1<br />
·<br />
S<br />
(q; q) i Nj n,q ,<br />
q<br />
s<br />
i=0<br />
On interchanging the order of summations and summing the resulting inner 0 Φ 0 (.)<br />
series with the help of a result Gasper and Rahman [7], namely,<br />
we obtain<br />
(q; q) ∞<br />
s 1+λ<br />
0Φ 0 (−; −; q, x) =<br />
[n/N]<br />
∑<br />
j=0<br />
j=0<br />
S n,q<br />
(q 1+kj+λ ; q) ∞<br />
·<br />
[<br />
n<br />
Nj<br />
1<br />
(x; q) ∞<br />
, (2.2)<br />
]<br />
q<br />
( 1<br />
s<br />
) kj<br />
,<br />
This, after certain simplifications reduces to the right hand side of (2.1).<br />
(1 − q) λ [n/N]<br />
∑<br />
[ ] ( ) kj<br />
n 1 − q<br />
s λ+1<br />
S<br />
Nj n,q Γ q (kj + λ + 1). (2.3)<br />
q<br />
s<br />
j=0<br />
3. Special Cases<br />
It is interesting to observe that in view of the definitions (1.23)-(1.26), the<br />
Theorem 2.1 leads to the q-Laplace transforms of the above mentioned polynomials<br />
after implementing the necessary changes in the values of S j,q , N and k. We<br />
illustrate the following cases:<br />
(i) If we take N = 1, k = 1 and S j,q = (q; q) 0 in Theorem 2.1, we obtain the<br />
q-Laplace transform of the q-Rogers-Szegö polynomial h n (x ; q) as<br />
{<br />
qL s x λ h n (x ; q) } = 1 ∑ n [ ]<br />
n<br />
s λ+1<br />
(q; q)<br />
j<br />
j (q 1+j ; q) λ (1/s) j . (3.1)<br />
q<br />
j=0<br />
□
86 R. K. YADAV, S. D. PUROHIT AND P. NIRWAN<br />
(ii) Again if we take N = 2, k = −2, λ = n + µ and S j,q = (q; q 2 ) j (−1) j q j(j−1) in<br />
the theorem (2.1), we obtained the q-Laplace transform of the discrete q-Hermite<br />
polynomial H n (x ; q) as:<br />
{<br />
qL s x µ+n H n (x ; q) } = (q; q) n/2<br />
∑<br />
n+µ<br />
s n+µ+1<br />
j=0<br />
(q −n ; q) 2j (−s 2 ) j q j(j−2µ−1)<br />
(q 2 ; q 2 ) j (q −µ ; q) 2j<br />
· (3.2)<br />
(iii) On setting N = 1, k = 1 and S j,q = (−1)n+j q −n(2n+1)<br />
2 +j 2 + j 2 (p; q) n<br />
in the<br />
(p; q) j<br />
main result (2.1), we obtain the q-Laplace transform of the Generalized Stieltjes-<br />
Weigert polynomial S n (x; p; q) as:<br />
{<br />
qL s x λ S n (x; p, q) } [<br />
= (−1)n q −n(2n+1)<br />
2 (p; q) n (q; q) λ q<br />
s λ+1 · −n , q 1+λ ;<br />
2Φ 2<br />
p , 0 ;<br />
q, − q s<br />
]<br />
n+ 3 2<br />
.<br />
(3.3)<br />
(iv) If we take N = 1, k = 1 and S j,q = (qα+n−1 ; q) j (2q) j q n(n−1)<br />
2 −nj+ j(j−1)<br />
2<br />
in the<br />
(−q; q) j<br />
Theorem 2.1, we obtain the q-Laplace transform of the q-Bessel function of second<br />
type y n (x; α/q 2 ) as<br />
qL s<br />
{<br />
x λ y n (x; α/q 2 ) } = q n(n−1)<br />
2 (q; q) λ<br />
s λ+1<br />
n ∑<br />
j=0<br />
(q −n ; q) j (q α+n−1 ; q) j (−2q/s) j (q λ+1 ; q) j<br />
(q; q) j (−q; q) j<br />
.<br />
(3.4)<br />
Similarly, one can deduce a number of known results due to Yadav and Purohit [15],<br />
involving the q-Laplace images of a variety of q-polynomials as the applications of<br />
the Theorem 2.1.<br />
4. q-Laplace Transforms of Basic Hypergeometric<br />
Functions and q-Polynomials<br />
In the following table, we enumerate the q-Laplace transforms of certain basic<br />
hypergeometric functions and q-polynomials. Some of the results deduced, are<br />
expressible in terms of the q-analogue of the Kampé-de Fériet functions.<br />
Eq.No. f(t) ϕ(s) ≡ q L s {f(t)} = 1<br />
(1−q)<br />
4.1 x λ ; λ > 0<br />
4.2 x ν e q (ax k ); k ∈<br />
4.3 e q (x) r Φ s<br />
[<br />
a1 , · · · , a r;<br />
b 1 , · · · , b s ;<br />
4.4 sin q(x) rΦ s<br />
[<br />
a1 , · · · , a r ;<br />
b 1 , · · · , b s ;<br />
4.5 cos q (x) r Φ s<br />
[<br />
a1 , · · · , a r ;<br />
b 1 , · · · , b s;<br />
]<br />
q, tx<br />
(q; q) ν<br />
s 1+ν<br />
∞ ∑<br />
] 1<br />
2is<br />
Φ 1 : 0 ; r<br />
q, tx<br />
]<br />
q, tx<br />
s∫<br />
−1<br />
0<br />
(q; q) λ<br />
s 1+λ<br />
E q (qst)f(t) d(t; q); Re(s) > 0<br />
(<br />
a/s k) r (q<br />
1+ν , q<br />
1+ν+1 , · · · , q<br />
1+ν+k−1 ; q<br />
k )r<br />
r=0<br />
(q; q) r<br />
1<br />
s<br />
Φ 1 : 0 ; r<br />
( )<br />
q : −; a1 , · · · , a r;<br />
q ;<br />
0 : 0 ; s − : − ; b 1 , · · · , b s ;<br />
1 s , s<br />
t ( )<br />
q : −; a1 , · · · , a r;<br />
q ;<br />
0 : 0 ; s − : − ; b 1 , · · · , b s ; s i , s<br />
t<br />
−<br />
2is 1 Φ 1 : 0 ; r<br />
( )<br />
q : −; a1 , · · · , a r ;<br />
q ; −i<br />
0 : 0 ; s − : − ; b 1 , · · · , b s ; s , s<br />
t<br />
1<br />
2s Φ 1 : 0 ; r<br />
(<br />
q : −; a1 , · · · , a r ;<br />
q ; i 0 : 0 ; s − : − ; b 1 , · · · , b s ; s , t )<br />
s<br />
1<br />
2s Φ 1 : 0 ; r<br />
(<br />
q : −; a1 , · · · , a r;<br />
q ; −i<br />
0 : 0 ; s − : − ; b 1 , · · · , b s; s , t )<br />
s
<strong>ON</strong> q-<strong>LAPLACE</strong> <strong>TRANSFORMS</strong> <strong>OF</strong> A <strong>GENERAL</strong> <strong>CLASS</strong>... 87<br />
1<br />
[ ]<br />
a1 , · · · , a<br />
4.6 sinh r ;<br />
q(x) rΦ s q, tx<br />
2s Φ 1 : 0 ; r<br />
(<br />
q : −; a1 , · · · , a r ;<br />
q ; 1 0 : 0 ; s − : − ; b 1 , · · · , b s ; s , t )<br />
s<br />
b 1 , · · · , b s ;<br />
−<br />
2s 1 Φ 1 : 0 ; r<br />
(<br />
q : −; a1 , · · · , a r;<br />
q ; −1<br />
0 : 0 ; s − : − ; b 1 , · · · , b s; s , t )<br />
s<br />
[ ] 1<br />
a1 , · · · , a<br />
4.7 cosh r ;<br />
2s<br />
Φ 1 : 0 ; r<br />
( )<br />
q : −; a1 , · · · , a r;<br />
q ;<br />
0 : 0 ; s − : − ; b<br />
q(x) rΦ s q, tx<br />
1 , · · · , b s ;<br />
1 s , s<br />
t<br />
b 1 , · · · , b s ;<br />
+ 1 2s Φ 1 : 0 ; r<br />
(<br />
q : −; a1 , · · · , a r ;<br />
q ; −1<br />
0 : 0 ; s − : − ; b 1 , · · · , b s ; s , t )<br />
[ ] s<br />
(q; q) ν (q; q)ν −;<br />
4.8 (x + a) ν<br />
s 1+ν or<br />
(−as; q) ∞ s 1+ν 0 Φ 0 q, −as<br />
−;<br />
[<br />
]<br />
4.9 x λ (q; q) ν+λ<br />
q −ν ;<br />
(x + a) ν ; λ > 0<br />
s 1+ν+λ 1 Φ 1<br />
q −ν−λ q , as/q<br />
;<br />
[<br />
4.10<br />
K Aff<br />
n (x; a, N; q)<br />
n > N<br />
4.11 J v (x ; q)<br />
4.12 J −v (x; q)<br />
4.13 P (α,β)<br />
n (x; γ, δ; q)<br />
4.14 R m,v (x ; q)<br />
(q; q) ∞<br />
s<br />
[<br />
1<br />
s 1+ν · 4Φ 4<br />
Φ 1 : 0 ; 2<br />
0 : 1 ; 2<br />
e ivπ [<br />
(q; q) −ν<br />
(q; q) ν s 1−ν · 4Φ 4<br />
1/s : − ; q −n , 0 ;<br />
− : 1/s ; aq, q −N ;<br />
q v+1<br />
2 , −q v+1<br />
2 , q v+2<br />
2 , −q v+2<br />
2 ;<br />
q v+1 , 0, 0, 0;<br />
q 1−ν<br />
2 , −q 1−ν<br />
2 , q 2−ν<br />
2 , −q 2−ν<br />
2 ;<br />
q 1−ν , 0 , 0 , 0;<br />
(αq, −δα q/γ, q) n (γ/α q) n (q; q) ∞<br />
(q, ⎡−q; q) n s<br />
qα<br />
1+ρ<br />
Φ 1 : 0; 2<br />
⎢ sγ : − ; q−n , αβ q n+1 ;<br />
0 : 1; 2 ⎣<br />
− : qα<br />
sγ<br />
−δαq<br />
; αq, ;<br />
γ<br />
(q; q) −m (q ν [<br />
; q) m q<br />
−n , q<br />
ν+m−n ;<br />
s 1−m 2Φ 1<br />
[<br />
q ν ;<br />
q 1−m<br />
2 , −q 1−m<br />
2 , q 2−m<br />
2 , −q 2−m<br />
2 ;<br />
4Φ 4<br />
q 1−ν−m , 0 , 0 , 0 ;<br />
]<br />
q, q, q<br />
q ,<br />
q, q 1+ρ , q⎥<br />
⎦<br />
q, q n+1 ]<br />
]<br />
q<br />
s 2<br />
]<br />
q, q−ν+1<br />
s 2<br />
⎤<br />
]<br />
q, q1−ν−m<br />
s 2<br />
To prove the result (4.2), we take f(x) = x ν e q (ax k ) in the equation (1.9) and<br />
make use the definition (1.4), which yields<br />
{<br />
qL s x ν e q (ax k ) } = (q; q) ∑ ∞<br />
∞ q j(1+ν) ∞<br />
{<br />
∑ a(s −1 q j ) k} r<br />
s 1+ν<br />
.<br />
(q; q) j (q, q) r<br />
j=0<br />
On interchanging the order of summations and then summing the resulting 0 Φ 0 (.)<br />
series with the help of equation (2.2), the right hand side of the above expression<br />
(q; q) ∞ (<br />
∞<br />
∑<br />
) a/s<br />
k r<br />
(q 1+ν , q 1+ν+1 , · · · , q 1+ν+k−1 ; q k ) r<br />
s 1+ν<br />
(4.15)<br />
(q; q)<br />
r=0<br />
r<br />
[ ]<br />
a1 , · · · , a<br />
For the proof of the result (4.3), we take f(x) = e q (x) r Φ r ;<br />
s q, tx in<br />
b 1 , · · · , b s ;<br />
the equation (1.9) and make use of definition (1.4) and (1.16), this yields;<br />
{ [ ]}<br />
a1 , · · · , a<br />
qL s e q (x) r Φ r ;<br />
s q, tx =<br />
b 1 , · · · , b s ;<br />
(q; q) ∞<br />
∞∑ q j ∑ ∞<br />
(s −1 q j ) r ∑ ∞<br />
(a 1 , · · · , a r ; q) k (ts −1 q j ) k<br />
s (q; q) j (q; q) r (q, b 1 , · · · , b s ; q) k<br />
j=0<br />
r=0<br />
On interchanging the order of summations and then summing the inner 0 Φ 0 (.)<br />
series with the help of equation (2.2), the right hand side of the above expression<br />
reduces<br />
(q; q) ∞<br />
s<br />
∞∑<br />
∞∑<br />
k=0 r=0<br />
k=0<br />
r=0<br />
(a 1 , · · · , a r ; q) k (t/s) k (1/s) r<br />
(q 1+r+k ; q) ∞ (q; q) r (q, b 1 , · · · , b s ; q) k<br />
(4.16)
<strong>88</strong> R. K. YADAV, S. D. PUROHIT AND P. NIRWAN<br />
On further simplification in the above expression we get the result (4.3). Proofs<br />
of the results (4.4) - (4.9) follow similarly.<br />
To prove the result (4.10), we take f(x) = Kn Aff (x; a, N; q) in the equation (1.9)<br />
and make use of the definition (1.27), which yields This further simplifies to the<br />
right hand side of the result (4.10).<br />
Proof of the results (4.11) - (4.14) follows similarly. We avoid the proofs for the<br />
sake of brevity.<br />
References<br />
[1] W. A. Abdi, On q-Laplace transforms, Proc. Nat. Acad. Sc. (India) 29(A) (1960), 389-407.<br />
[2] R. P. Agarwal, Certain Fractional q-integrals and q-derivatives, Proc. Camb. Phil. Soc. 66<br />
(1969), 365-370.<br />
[3] W. A. Al-Salam, Some Fractional q-integrals and q-derivatives, Proc. Edin. Math. Soc. 15<br />
(1966), 135-140.<br />
[4] G. E. Andrews, R. Askey and Ranjan Roy, Special Functions. Cambridge University Press,<br />
Cambridge, 1999.<br />
[5] N.M. Atakishiyev and Sh. M. Nagiyev, On the Rogers Szegö Polynomials, J. Physics -A, 27<br />
(1994), L611-L615.<br />
[6] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions,<br />
Vol. I, McGraw-Hill Inc. New York, 1953.<br />
[7] G. Gasper, and M. Rahman, Basic Hypergeometric Series, Cambridge University Press,<br />
Cambridge, 1990.<br />
[8] W. Hahn, Beitrage Zur Theorie der Heineschen Rèihen die 24 integral der. hypergeometrischen<br />
q-differenzeng Leichung, das q-Analogon der Laplace-Transformation, Math.<br />
Nachr, 2 (1949), 340-379.<br />
[9] V. K. Jain and H. M. Srivastava, Some families of Multilinear q-Generating functions and<br />
Combinatorial q-Series Identities, J. Math. Anal. Appl. 192(2) (1995), 413-438.<br />
[10] H. T. Koelink and R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel function and<br />
Associated q-Lommel Polynomials, Journal of Mathematical Anal. and Math. Appl. Vol. 186<br />
(1994), 690-710.<br />
[11] I. N. Sneddon, The use of integral transforms, Tata McGraw-Hill Publishing, Co. Ltd., New<br />
Delhi, 1974.<br />
[12] H. M. Srivastava and A. K. Agarwal, Generating functions for a class of q- Polynomials,<br />
Annali di Matematica Pura ed Applicata, 154(1) (1989), 99-109.<br />
[13] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, John Wiley<br />
and Sons, Halsted press, New York, 1985.<br />
[14] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge,<br />
1966.<br />
[15] R. K. Yadav, S. D. Purohit, On q-Laplace transforms of Certain q-hypergeometric polynomials,<br />
Proc. Nat. Acad. Sci. India 76(A) III (2006), 235-242.<br />
[16] R. K. Yadav, S. D. Purohit, On q-Laplace transforms of certain multiple basic hypergeometric<br />
functions, Math. Student. 74 (2005), 207-215.<br />
[17] R. K. Yadav, S. D. Purohit, On q-Laplace transforms of certain generalized basic hypergeometric<br />
functions, Proc. 5th International Conf. of SSFA 5 (2004), 74-81.<br />
∗ Department of Mathematics and Statistics,<br />
J. N. V. University, Jodhpur-342 005, India.<br />
E-mail address: rkmdyadav@yahoo.co.in<br />
∗∗ Department of basic science (Mathematics),<br />
College of Technology and Engineering,<br />
M.P University of Agriculture and Technology, Udaipur, India.<br />
E-mail address: sunil a purohit@yahoo.com