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Applied Mathematics, 2019, 10, 301-311
http://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385
DOI:
10.4236/am.2019.105021 May 7, 2019 301 Applied Mathematics
On q-Analogues of Laplace Type Integral
Transforms of q
2
-Bessel Functions
Arwa Al-Shibani, R. T. Al-Khairy
Department of Mathematics, Faculty of Sciences, Imam Abdulrahman Bin Faisal University, Dammam, KSA
Abstract
The present paper deals with the evaluation of the
q
-
Analogues of Laplece
transforms of a product of basic analogues of
q
2
-
special functions. We apply
these transforms to three families of
q
-
Bessel functions. Several special cases
have been deducted.
Keywords
q
-Extensions of Bessel Functions,
q
-Analogues of Laplace Type Integrals
Transforms,
q
-Analogues of Gamma Function,
q
-Shift Factorials
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
( )
( )
( ) ( )
2
1
0
4
; ,2
2,;
n
n
nn
z
z
J zq z
qq qq
µ
µ
µ
∞
=
+
−
= <
∑
(1)
( )
( )
( )
( ) ( )
2
2
0
4
;,
2,;
n
nn
n
nn
z
q
z
J zq z
qq qq
µ
µ
µ
µ
+
∞
=
+
−
= ∈
∑
(2)
How to cite this paper:
Al-Shibani,
A. and
Al
-Khairy, R.T. (2019) On
q
-
Analogues of
Laplace Type Integral Transforms of
q
2
-
Bessel
Functions
.
Applied
Mathematics
,
10
, 301-311.
https://doi.org/10.4236/am.2019.105021
Received:
March 11, 2019
Accepted:
May 4, 2019
Published:
May 7, 2019
Copyright © 201
9 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the
Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
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