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On q-Analogues of Laplace Type Integral Transforms of q2-Bessel Functions

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DOI: 10.4236/am.2019.105021    224 Downloads   403 Views  

ABSTRACT

The present paper deals with the evaluation of the q-Analogues of Laplece transforms of a product of basic analogues of q2-special functions. We apply these transforms to three families of q-Bessel functions. Several special cases have been deducted.

1. Introduction

In the second half of twentieth century, there was a significant increase of activity in the area of the q-calculus mainly due to its application in mathematics, statistics and physics. In literature, several aspects of q-calculus were given to enlighten the strong inter disciplinary as well as mathematical character of this subject. Specifically, there have been many q-analogues and q-series representations of various kinds of special functions. In the case of q-Bessel function, there are two related q-Bessel functions introduced by Jackson [1] and denoted by Ismail [2] as

J μ ( 1 ) ( z ; q ) = ( z 2 ) μ n = 0 ( z 2 4 ) n ( q , q ) μ + n ( q ; q ) n , | z | < 2 (1)

J μ ( 2 ) ( z ; q ) = ( z 2 ) μ n = 0 q n ( n + μ ) ( z 2 4 ) n ( q , q ) μ + n ( q ; q ) n , z (2)

The third related q-Bessel function J μ ( 3 ) ( z ; q ) was introduced in a full case as [3]

J μ ( 3 ) ( z ; q ) = z μ n = 0 ( 1 ) n q n ( n 1 ) 2 ( q z 2 ) n ( q , q ) μ + n ( q ; q ) n , z (3)

A certain type of Laplace transforms, which is called L2-transform, was introduced by Yürekli and Sadek [4] . Then these transforms were studied in more details by Yürekli [5] , [6] . Purohit and Kalla applied the q-Laplace transforms to a product of basic analogues of the Bessel function [7] .

On the same manner, integral transforms have different q-analogues in the theory of q-calculus. The q-analogue of the Laplace type integral of the first kind is defined by [8] as

q L 2 ( f ( ξ ) ; y ) = 1 1 q 2 0 y 1 ξ E q 2 ( q 2 y 2 ξ 2 ) f ( ξ ) d ξ (4)

and expressed in terms of series representation as

q L 2 ( f ( ξ ) ; y ) = ( q 2 ; q 2 ) [ 2 ] q y 2 i = 0 q 2 i ( q 2 ; q 2 ) i f ( q i y 1 ) . (5)

On the other hand, the q-analogue of the Laplace type integral of the second kind is defined by [8] as

q l 2 ( f ( ξ ) ; y ) = 1 1 q 2 0 ξ e q 2 ( y 2 ξ 2 ) f ( ξ ) d q ξ (6)

whose q-series representation expressed as

q l 2 ( f ( ξ ) ; y ) = 1 [ 2 ] 2 ( y 2 ; q 2 ) i q 2 i f ( q i ) ( y 2 ; q 2 ) i . (7)

In this paper we build upon analysis of [8] . Following [9] , we discuss the q-Laplace type integral transforms (4) and (7) on the q-Bessel functions J μ ( 1 ) ( z ; q ) , J μ ( 2 ) ( z ; q ) and J μ ( 3 ) ( z ; q ) , respectively. In Section 2, we recall some notions and definitions from the q-calculus. In Section 3, we give the main results to evaluate the q-analogue of Laplace transformation of q2-Basel function. In Section 4, we discuss some special cases.

2. Definitions and Preliminaries

In this section, we recall some usual notions and notations used in the q-theory. It is assumed in this paper wherever it appears that 0 < q < 1 . For a complex number a, the q-analogue of a is introduced as [ a ] q = 1 q a 1 q . Also, by fixing a , the q-shifted factorials are defined as

( a ; q ) 0 = 1 ; ( a , q ) n = k = 0 n 1 ( 1 a q k ) , n = 1,2, ; ( a ; q ) = lim n ( a ; q ) n . (8)

This indeed lead to the conclusion

( [ n ] q ) ! = ( q ; q ) n ( 1 q ) n , n and ( a ; q ) x = ( a ; q ) ( a q x ; q ) . (9)

The q-analogue of the exponential function of first and second type are respectively given in [10] by

e q ( x ) = 0 x n ( q ; q ) n = 1 ( x ; q ) , | x | < 1. (10)

and

E q ( x ) = 0 ( 1 ) n q n n 1 2 x n ( q ; q ) n , x . (11)

Indeed it has been shown that

e q ( x ) = 1 ( x ; q ) , | x | < 1 and E q ( x ) = ( x , q ) , x (12)

The finite q-Jackson and improper integrals are respectively defined by [11]

0 x f ( t ) d q t = x ( 1 q ) k = 0 q k f ( x q k ) (13)

and

0 / A f ( t ) d q t = ( 1 q ) k q k A f ( q k A ) . (14)

The q-analogues of the gamma function of first and second type are respectively defined in [9] as

Γ q ( α ) = 0 1 / ( 1 q ) x α 1 E q ( q ( 1 q ) x ) d q x , ( α > 0 ) (15)

and

q Γ ( α ) = K ( A ; α ) 0 / A ( 1 q ) x α 1 e q ( ( 1 q ) x ) d q x (16)

where, α 1 > 0 , where K ( A ; α ) is the function given by

K ( A ; α ) = A α 1 ( q / α ; q ) ( α ; q ) ( q t / α ; q ) ( α q 1 t ; q ) . (17)

Some useful results, for x 0, 1, 2, , we use here are given by

Γ q ( α ) = ( q ; q ) ( 1 q ) α 1 k = 0 q k α ( q ; q ) k = ( q ; q ) ( q α ; q ) ( 1 q ) 1 x , (18)

and

q Γ ( α ) = K ( A ; α ) ( 1 q ) α 1 ( 1 A ; q ) k ( q k A ) ( 1 A ; q ) k . (19)

3. Main Theorems

Theorem 1. Let J 2 μ 1 ( 1 ) ( 2 a 1 t ; q 2 ) , , J 2 μ n ( 1 ) ( 2 a n t ; q 2 ) be a set of first kind of q2-Bessel functions, f ( t ) = t Δ 1 j = 1 n J 2 μ j ( 1 ) ( 2 a j t ; q 2 ) , where Δ , a j and μ j for j = 1 , 2 , , n are constants; then the q-analogue of Lablace transformation q L 2 of f ( t ) is given as:

q L 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( a j s ) m j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 ) (20)

and the q-analogue of Laplace transformation q l 2 of f ( t ) is given as:

q l 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( a j s ) m j B m j ( q 2 ) q 2 Γ ( m j + μ j + Δ + 1 2 ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) . (21)

where R e ( s ) > 0 , R e ( Δ ) > 0 and

A Δ = ( 1 q 2 ) Δ / 2 [ 2 ] s Δ + 1 ( q 2 ; q 2 ) , B m j ( q 2 ) = ( q 2 μ j + m j + 2 ; q 2 ) ( 1 q 2 ) m j + μ j 1 2 ( q 2 ; q 2 ) m j

Proof. Now,

q L 2 ( f ( t ) ; s ) = ( q 2 ; q 2 ) [ 2 ] s 2 k = 0 q 2 k f ( q k s 1 ) ( q 2 ; q 2 ) k

since

J 2 μ j ( 1 ) ( 2 a j t ; q 2 ) = ( 2 a j t 2 ) 2 μ j m j = 0 ( ( 2 a j t ) 2 4 ) m j ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j

so

q L a ( f ( t ) ; s ) = ( q 2 ; q 2 ) [ 2 ] s 2 k = 0 q 2 k ( q 2 ; q 2 ) k ( q k s 1 ) Δ 1 j = 1 n ( a j q k s 1 ) 2 μ j m j = 0 ( 1 ) m j ( a j q k s 1 ) m j ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j = ( q 2 ; q 2 ) [ 2 ] s Δ + 1 k = 0 q k ( Δ + 1 ) ( q 2 ; q 2 ) k j = 1 n ( a j q k s ) μ j m j = 0 ( 1 ) m j ( a j q k s ) m j ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j = ( q 2 ; q 2 ) [ 2 ] s Δ + 1 j = 1 n ( a j s ) μ j m j = 0 ( 1 ) m j ( a j s ) m j ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j k = 0 q k ( Δ + 1 + m j + μ j ) ( q 2 ; q 2 ) k (22)

Since

Γ q 2 ( α ) = ( q 2 ; q 2 ) ( 1 q 2 ) α 1 k = 0 q 2 k α ( q 2 ; q 2 ) k

putting α = 1 + Δ + m j + μ j 2 , so (22) becomes:

q L s ( f ( t ) ; s ) = 1 [ 2 ] s Δ + 1 j = 1 n ( a j s ) μ j m j = 0 ( 1 ) m j ( a j s ) m j ( 1 q 2 ) 1 + Δ + m j + μ j 2 ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j .

Γ q 2 ( m j + μ j + Δ + 1 2 ) (23)

Since

( q 2 ; q 2 ) 2 μ j + m j = ( q 2 ; q 2 ) ( q 2 q 2 μ j + m j ; q 2 )

so (23) becomes:

q L 2 ( f ( t ) ; s ) = ( 1 q 2 ) Δ / 2 [ 2 ] s Δ + 1 ( q 2 ; q 2 ) j = 1 n ( a j s ) μ j m j = 0 ( a j s ) ( 1 ) m j ( q 2 μ j + m j + 2 ; q 2 ) ( 1 q 2 ) m j + μ j 1 2 ( q 2 ; q 2 ) m j .

Γ q 2 ( m j + μ j + Δ + 1 2 ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( a j s ) ( 1 ) m j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 ) 2

Similarly we have

q l 2 ( f ( t ) ; s ) = 1 [ 2 ] 1 ( s 2 ; q 2 ) k = 0 q 2 k ( s 2 ; q 2 ) k ( q k ) Δ 1 j = 1 n J 2 μ j ( 1 ) ( 2 a j q k ; q 2 ) = 1 [ 2 ] 1 ( s 2 ; q 2 ) k = 0 q 2 k ( s 2 ; q 2 ) k ( q k ) Δ 1 j = 1 n ( a j q k ) μ j .

m j = 0 ( a j q k ) m j ( q 2 ; q 2 ) m j + 2 μ j = j = 1 n ( a j ) μ j [ 2 ] m j = 0 ( a j ) m j ( q 2 ; q 2 ) m j + 2 μ j ( q 2 ; q 2 ) m j k = 0 ( s 2 ; q 2 ) k q k ( m j + μ j + Δ + 1 ) ( s 2 ; q 2 )

Now using

q 2 Γ ( α ) = K ( A ; α ) ( 1 q 2 ) α 1 ( 1 A ; q 2 ) k Z ( q K A ) α ( 1 A ; q 2 ) K

with A = 1 s 2 , α = m j + μ j + Δ + 1 2 we get

q l 2 ( f ( t ) ; s ) = j = 1 m ( a j ) μ j [ 2 ] s μ j + Δ + 1 m j = 0 ( a j s ) m j ( 1 q 2 ) m j + μ j + Δ + 1 2 q 2 Γ ( m j + μ j + Δ + 1 2 ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) ( q 2 ; q 2 ) m j + 2 μ j ( q 2 ; q 2 ) m j = ( 1 q 2 ) Δ 2 [ 2 ] s Δ + 1 ( q 2 ; q 2 ) j = 1 m ( a j s ) μ j m j = 0 ( a j s ) m j ( 1 q 2 ) m j + μ j 1 2 ( q m j + 2 μ j + 2 ; q 2 ) K ( 1 s 2 , m j + μ j + Δ + 1 2 ) ( q 2 ; q 2 ) m j .

q 2 Γ ( m j + μ j + Δ + 1 2 ) = A Δ j = 1 m ( a j s ) μ j m j = 0 ( a j s ) m j K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 )

Theorem 2. Let J 2 μ 1 ( 2 ) ( 2 a , t ; q 2 ) , , J 2 μ n ( 2 ) ( 2 a t ; q 2 ) be a set of second order q2-Bessel function, f ( t ) = t Δ 1 j = 1 n J 2 μ j ( 2 ) ( 2 a j t ; q 2 ) where Δ , a j and μ j for j = 1 , 2 , , n are constants then q L 2 -transform of f ( t ) is given as:

q L 2 ( f ( t ) , s ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( 1 ) m j q 2 m j ( m j + 2 μ j ) ( a j s ) m j + μ j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 ) (24)

and the q-analogue of Laplace transformation q l 2 of f ( t ) is given as:

q l 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( a j s ) m j q 2 m j ( m j + 2 μ j ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 ) (25)

Proof. Now,

J 2 μ j ( 2 ) ( 2 a j t ; q 2 ) = ( 2 a j t 2 ) 2 μ j m j ( ( 2 a j t ) 2 4 ) m j q 2 m j ( m j + 2 a j ) ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j

so

q L 2 ( f ( t ) ; s ) = ( q 2 ; q 2 ) [ 2 ] s 2 k = 0 q 2 k ( q 2 ; q 2 ) k ( q k s 1 ) Δ 1 j = 1 n ( 2 a j q k s 1 2 ) 2 μ j m j = 0 ( ( 2 a j q k s 1 ) 2 4 ) m j q 2 m j ( m j + 2 μ j ) ( q 2 ; q 2 ) 2 μ j + m j ( q 2 ; q 2 ) m j (26)

By the same argument we can write (26) as

q L 2 ( f ( t ) ; s ) = ( q 2 ; q 2 ) [ 2 ] s Δ + 1 ( q 2 ; q 2 ) j = 1 n m j = 0 ( 1 ) m j q 2 m j ( m j + 2 μ j ) ( q 2 ; q 2 ) m j ( a j s ) m j + μ j ( q 2 μ j + m j + 2 ; q 2 ) k = 0 q k ( m j + μ j + 1 + Δ ) ( q 2 ; q 2 ) k

put α = m j + μ j + Δ + 1 2 in Γ q 2 ( α ) , then

So (25) becomes:

q L 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j s ) μ j m j = 0 ( 1 ) m j q 2 m j ( m j + 2 μ j ) ( a j s ) m j + μ j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 )

Similarly

q l 2 ( f ( t ) ; s ) = 1 [ 2 ] 1 ( s 2 ; q 2 ) k = 0 q 2 k ( s 2 ; q 2 ) k ( q k ) Δ 1 j = 1 n ( a j q k ) μ j m j = 0 ( a j q k ) m j q 2 m j ( m j + 2 μ j ) ( q 2 ; q 2 ) m j + 2 μ j ( q 2 ; q 2 ) m j

Put A = 1 s 2 , α = m j + μ j + Δ + 1 2 we get

q l 2 ( f ( t ) ; s ) = 1 [ 2 ] j = 1 n ( a j ) μ j m j = 0 ( a j ) m j q 2 m j ( m j + 2 μ j ) ( 1 q 2 ) m j + μ j + Δ + 1 2 q 2 Γ ( m j + μ j + Δ + 1 2 ) ( q 2 ; q 2 ) m j + 2 μ j K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) s m j + μ j + Δ + 1 = A Δ m j = 0 ( a j s ) m j q 2 m j ( m j + 2 μ j ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 )

Theorem 3. Let J 2 μ j ( 3 ) ( q 1 a 1 t ; q 2 ) , , J 2 μ n ( 3 ) ( q 1 a n t ; q 2 ) be s set of q2-Bessel functions, f ( t ) = t Δ 1 j = 1 n J 2 μ j ( 3 ) ( q 1 a j t ; q 2 ) where Δ , a j and μ j for j = 1 , 2 , , n are constants. Then we have

q L 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j q s ) μ j m j = 0 ( 1 ) m j q m j ( m j 1 ) ( a j q s ) m j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 ) (27)

and the q-analogue of Laplace transformation q l 2 of f ( t ) is given by:

q l 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j q s ) μ j m j = 0 ( a j q s ) m j q m j ( m j 1 ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 ) 2 (28)

Proof. Now

J 2 μ j ( 3 ) ( a j q k 1 s 1 ; q 2 ) = ( a j q k 1 s 1 ) 2 μ j m j = 0 ( 1 ) m j q 2 m j ( m j 1 ) 2 ( q 2 a j q k 1 s 1 ) m j ( q 2 ; q 2 ) m j + 2 μ j ( q 2 ; q 2 ) m j

q L 2 ( f ( t ) ; s ) = ( q 2 ; q 2 ) [ 2 ] s 2 k = 0 q 2 k ( q k s 1 ) Δ 1 ( q 2 ; q 2 ) k j = 1 n ( a j q k 1 s 1 ) μ j m j = 0 ( 1 ) m j q m j ( m j 1 ) ( q 2 a j q k 1 s 1 ) m j ( q 2 ; q 2 ) m j + 2 μ j ( q 2 ; q 2 ) m j

put α = m j + μ j + Δ + 1 2 , we get

q L 2 ( f ( t ) ; s ) = A Δ j = 1 n ( a j q s ) μ j m j = 0 ( 1 ) m j q m j ( m j 1 ) ( a j q s ) m j B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 )

Similarly

q l 2 ( f ( t ) ; s ) = 1 [ 2 ] 1 ( s 2 ; q 2 ) j = 1 n ( q k 1 ) μ j ( a j ) μ j m j = 0 ( 1 ) m j q m j ( m j 1 ) ( q k m j ) ( q a j ) m j ( q 2 ; q 2 ) m j + μ 2 ( q 2 ; q 2 ) m j k = 0 q k ( Δ + 1 ) ( s 2 ; q 2 ) k .

Put α = m j + μ j + Δ + 1 2 , A = 1 s 2 we get

q l 2 ( f ( t ) ; s ) = ( 1 q 2 ) Δ 2 [ 2 ] s Δ + 1 ( q 2 ; q 2 ) j = 1 n ( a j q s ) μ j m j = 0 ( a j q s ) m j q m j ( m j 1 ) B m j ( q 2 ) Γ q 2 ( m j + μ j + Δ + 1 2 ) K ( 1 s 2 ; m j + μ j + Δ + 1 2 ) .

4. Special Cases

1) Let n = 1 , μ 1 = μ , a 1 = a in above theorems, respectively we have:

q L 2 ( t Δ 1 J 2 μ ( 1 ) ( 2 a t ; q 2 ) ; s ) = A Δ ( a s ) μ m = 0 ( a s ) m B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (29)

q l 2 ( t Δ 1 J 2 μ ( 1 ) ( 2 a t ; q 2 ) ; s ) = A Δ ( a s ) μ m = 0 ( a s ) m K ( 1 s 2 ; m + μ + Δ + 1 2 ) B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (30)

q L 2 ( t Δ 1 J 2 μ ( 2 ) ( 2 a t ; q 2 ) ; s ) = A Δ ( a s ) μ m = 0 ( 1 ) m q 2 m ( m + 2 μ ) B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (31)

q l 2 ( t Δ 1 J 2 μ ( 2 ) ( 2 a t ; q 2 ) ; s ) = A Δ ( a s ) μ m = 0 ( a s ) m q 2 m ( m + 2 μ ) K ( 1 s 2 ; m + μ + Δ + 1 2 ) B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (32)

q L 2 ( t Δ 1 J 2 μ ( 3 ) ( 2 a q 1 t ; q 2 ) ; s ) = A Δ ( a q s ) μ m = 0 ( 1 ) m q m ( m 1 ) ( a q s ) m B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (33)

q l 2 ( t Δ 1 J 2 μ ( 3 ) ( 2 a q 1 t ; q 2 ) ; s ) = A Δ ( a s ) μ m = 0 ( a q s ) m q m ( m 1 ) K ( 1 s 2 ; m + μ + Δ + 1 2 ) B m ( q 2 ) Γ q 2 ( m + μ + Δ + 1 2 ) (34)

2) Put Δ 1 = μ in part (29) above, then

q L 2 ( t μ J 2 μ ( 1 ) ( 2 a t ; q 2 ) ; s ) = ( 1 q 2 ) μ + 1 2 [ 2 ] s μ + 2 ( q 2 ; q 2 ) ( a s ) μ

m = 0 ( a s ) m ( q 2 μ + m + 2 ; q 2 ) ( 1 q 2 ) m + μ 1 2 ( q 2 ; q 2 ) m Γ q 2 ( m + 2 μ + 2 2 ) = ( a s ) μ [ 2 ] s μ + 2 m = 0 ( a s ) m ( q 2 ; q 2 ) m = ( a ) μ [ 2 ] s 2 μ + 2 e q 2 ( a s ) .

3) Put μ = 0 we get

q L 2 ( J 0 ( 1 ) ( 2 a t ; q 2 ) ; s ) = 1 [ 2 ] s 2 e q 2 ( a s ) .

which is the same result cited by [7] .

4) Put Δ 1 in (33), then

q L 2 ( t μ J 2 μ ( 3 ) ( 2 q 1 a t ) ; s ) = ( 1 q 2 ) μ + 1 2 [ 2 ] s μ + 2 ( q 2 ; q 2 ) ( a q s ) μ .

m = 0 ( 1 ) m q m ( m 1 ) ( a q s ) m ( q 2 μ + m + 2 ; q 2 ) ( 1 q 2 ) m + μ 1 2 Γ q 2 ( m + 2 μ + 2 2 ) ( q 2 ; q 2 ) m = ( a q ) μ [ 2 ] s 2 μ + 2 m = 0 ( 1 ) m ( a q s ) m q 2 m m 1 2 ( q 2 ; q 2 ) m = ( a q ) μ [ 2 ] s 2 μ + 2 E q 2 ( a q s ) .

5) Let μ = 0 and a = 0 in (34), then

q L 2 ( t Δ 1 ; s ) = ( 1 q 2 ) Δ 2 [ 2 ] s Δ + 1 1 K ( 1 s 2 ; Δ + 1 2 ) ( 1 q 2 ) 1 2 Γ q 2 ( Δ + 1 2 )

replacing Δ 1 by α , we get

q L 2 ( t α ; s ) = ( 1 q 2 ) α 2 [ 2 ] s α + 2 1 K ( 1 s 2 ; 1 + α 2 ) Γ q 2 ( 1 + α 2 )

which is the same result in [8] .

Acknowledgements

The authors are thankful to Professor S. K. Al-Omari for his suggestions in this paper.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Alshibani, A. and Alkhairy, R. (2019) On q-Analogues of Laplace Type Integral Transforms of q2-Bessel Functions. Applied Mathematics, 10, 301-311. doi: 10.4236/am.2019.105021.

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