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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

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