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
(1)
(2)
The third related q-Bessel function
was introduced in a full case as [3]
(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
(4)
and expressed in terms of series representation as
(5)
On the other hand, the q-analogue of the Laplace type integral of the second kind is defined by [8] as
(6)
whose q-series representation expressed as
(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
,
and
, 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
. For a complex number a, the q-analogue of a is introduced as
. Also, by fixing
, the q-shifted factorials are defined as
(8)
This indeed lead to the conclusion
(9)
The q-analogue of the exponential function of first and second type are respectively given in [10] by
(10)
and
(11)
Indeed it has been shown that
(12)
The finite q-Jackson and improper integrals are respectively defined by [11]
(13)
and
(14)
The q-analogues of the gamma function of first and second type are respectively defined in [9] as
(15)
and
(16)
where,
, where
is the function given by
(17)
Some useful results, for
, we use here are given by
(18)
and
(19)
3. Main Theorems
Theorem 1. Let
be a set of first kind of q2-Bessel functions,
, where
,
and
for
are constants; then the q-analogue of Lablace transformation
of
is given as:
(20)
and the q-analogue of Laplace transformation
of
is given as:
(21)
where
,
and
Proof. Now,
since
so
(22)
Since
putting
, so (22) becomes:
(23)
Since
so (23) becomes:
Similarly we have
Now using
with
,
we get
Theorem 2. Let
be a set of second order q2-Bessel function,
where
and
for
are constants then
-transform of
is given as:
(24)
and the q-analogue of Laplace transformation
of
is given as:
(25)
Proof. Now,
so
(26)
By the same argument we can write (26) as
put
in
, then
So (25) becomes:
Similarly
Put
we get
Theorem 3. Let
be s set of q2-Bessel functions,
where
and
for
are constants. Then we have
(27)
and the q-analogue of Laplace transformation
of
is given by:
(28)
Proof. Now
put
, we get
Similarly
Put
,
we get
4. Special Cases
1) Let
,
,
in above theorems, respectively we have:
(29)
(30)
(31)
(32)
(33)
(34)
2) Put
in part (29) above, then
3) Put
we get
which is the same result cited by [7] .
4) Put
in (33), then
5) Let
and
in (34), then
replacing
by
, we get
which is the same result in [8] .
Acknowledgements
The authors are thankful to Professor S. K. Al-Omari for his suggestions in this paper.