mn> 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.

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.
References

[1]   Jackson, F.H. (1905) The Application of Basic Numbers to Bessel’s and Legendre’s Functions. Proceedings of the London Mathematical Society, 2, 192-220.
https://doi.org/10.1112/plms/s2-2.1.192

[2]   Ismail, M.E.H. (1982) The Zeros of Basic Bessel Function, the Functions Jv+ax(x), and Associated Orthogonal Polynomials. Journal of Mathematical Analysis and Applications, 86, 1-19.
https://doi.org/10.1016/0022-247X(82)90248-7

[3]   Exton, H. (1978) A Basic Analogue of the Bessel-Clifford Equation. Jnanabha, 8, 49-56.

[4]   Yürekli, O. and Sadek, I. (1991) A Parseval-Goldstein Type Theorem on the Widder Potential Transform and Its Applications. International Journal of Mathematics and Mathematical Sciences, 14, 517-524.
https://doi.org/10.1155/S0161171291000704

[5]   Yürelki, O. (1999) Theorems on L2-Transforms and Its Application. Complex Variables, Theory and Application: An International Journal, 38, 95-107.
https://doi.org/10.1080/17476939908815157

[6]   Yürekli, O. (1999) New Identities Involving the Laplace and the L2-Transforms and Their Applications. Applied Mathematics and Computation, 99, 141-151.
https://doi.org/10.1016/S0096-3003(98)00002-2

[7]   Purohit, S.D. and Kalla, S.L. (2007) On q-Laplace Transforms of the q-Bessel Functions. Fractional Calculus and Applied Analysis, 10, 189-196.

[8]   Uçar, F. and Albayrak, D. (2011) On q-Laplace Type Integral Operators and Their Applications. Journal of Difference Equations and Applications, 18, 1001-1014.

[9]   Al-Omari, S.K.Q. (2017) On q-Analogues of the Natural Transform of Certain q-Bessel Function and Some Application. Filomat, 31, 2587-2598.
https://doi.org/10.2298/FIL1709587A

[10]   Hahn, W. (1949) Beitrage Zur Theorie der Heineschen Reihen, Die 24 Integrale der hypergeometrischen q-Differenzengleichung, Das q-Analogon der Laplace Transformation. Mathematische Na-chrichten, 2, 340-379.
https://doi.org/10.1002/mana.19490020604

[11]   Kac, V.G. and De Sele, A. (2005) On Integral Representations of q-Gamma and q-Beta Functions. Accademia Nazionale dei Lincei, 16, 11-29.

 
 
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