Advances in Difference Equations

, 2018:124 | Cite as

Some results for Laplace-type integral operator in quantum calculus

  • Shrideh K. Q. Al-Omari
  • Dumitru Baleanu
  • Sunil D. Purohit
Open Access
Research
  • 160 Downloads

Abstract

In the present article, we wish to discuss q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox’s \(H_{q}\)-functions. Some of the discussed functions are the q-Bessel functions of the first kind, the q-Bessel functions of the second kind, the q-Bessel functions of the third kind, and the q-Struve functions as well. Also, we obtain some associated results related to q-analogues of the Laplace-type integral on hyperbolic sine (cosine) functions and some others of exponential order type as an application to the given theory.

Keywords

\(J_{v}(x;q)\) function \(Y_{v}(x;q)\) function \(K_{v}(x;q)\) function \(H_{v}(x;q)\) function Laplace-type integral 

1 Introduction and preliminaries

Quantum calculus is a version of calculus where derivatives are differences and antiderivatives are sums, and no further limits are required. The quantum calculus or q-calculus, compared to the differential and integral calculus, has been very recently named. Hence some rules and definitions need to be recalled. For \(0< q<1\), the q-calculus starts with the definition of the q-analogue of the differential and the q-analogue of derivatives as well. The q-analogue of the integer n, the factorial of n, and the binomial coefficient are respectively given as
$$ [ n ] _{q}=\frac{1-q^{n}}{1-q}, \quad\quad \bigl( [ n ] _{q} \bigr) != \left \{ \textstyle\begin{array}{l@{\quad}l} \prod_{1}^{n} [ k ] _{q} , & n\in \mathbb{N} \\ 1 , & n=0 \end{array}\displaystyle \right \} , \quad\quad \left [ \begin{matrix} n \\ k \end{matrix} \right ] _{q}=\prod_{1}^{n} \frac{1-q^{n-k+1}}{1-q^{k}}. $$
(1)
The q-analogue of \(( x+a ) ^{n}\) (\(n\in \mathbb{N} \)) and its q-derivative are respectively given as
$$ ( x+a ) _{q}^{n}=\prod_{j=0}^{n-1} \bigl( x+q^{j}a \bigr) , \quad \quad D_{q} ( x+a ) _{q}^{n}= [ n ] _{q} ( x+a ) _{q}^{n-1},\quad \quad ( x+a ) _{q}^{0}=1. $$
(2)
The q-Jackson integrals from 0 to a and from a to b are given as follows (see [1], see also [2]):
$$ \int _{0}^{a}f ( x ) \,d_{q}x= ( 1-q ) a\sum _{0}^{\infty }f \bigl( aq^{k} \bigr) q^{k} $$
(3)
and
$$ \int _{b}^{a}f ( x ) \,d_{q}x= \int _{0}^{b}f ( x ) \,d_{q}x- \int _{0}^{a}f ( x ) \,d_{q}x. $$
(4)
The improper q-Jackson integral is given as follows (see [1]):
$$ \int _{0}^{\frac{\infty }{A}}f ( x ) \,d_{q}x= ( 1-q ) \sum _{n\in \mathbb{Z} }\frac{q^{k}}{A}f \biggl( \frac{q^{k}}{A} \biggr) , \quad A\in \mathbb{C}. $$
The q-analogues of the gamma function are defined by
$$ \Gamma_{q} ( \alpha ) = \int _{0}^{\frac{1}{1-q}}x ^{\alpha -1}E_{q} \bigl( q ( 1-q ) x \bigr) \,d_{q}x $$
and
$$ _{q}\Gamma ( \alpha ) =K ( A;\alpha ) \int _{0}^{\frac{\infty }{A ( 1-q ) }}x^{\alpha -1}e_{q} \bigl( - ( 1-q ) x \bigr) \,d_{q}x, $$
where \(\alpha >0\) and, for every \(t\in \mathbb{R}\),
$$ K ( A;t ) =A^{t-1}\frac{ ( -q/A;q ) _{ \infty }}{ ( -q^{t}/A;q ) _{\infty }}\frac{ ( -A;q ) _{\infty }}{ ( -Aq^{1-t};q ) _{\infty }}. $$
Here
$$ ( a;q ) _{n}=\prod_{0}^{n-1} \bigl( 1-aq^{k} \bigr) , \quad\quad ( a;q ) _{\infty }= \stackrel{\lim }{n \rightarrow \infty } ( a;q ) _{n}. $$
The very useful identities used in this article are (cf. [2])
$$ \Gamma_{q} ( x ) =\frac{ ( q;q ) _{\infty }}{ ( q^{x};q ) _{\infty }} ( 1-q ) ^{1-x}\quad \text{and}\quad ( a;q ) _{t}=\frac{ ( a;q ) _{\infty }}{ ( aq^{t};q ) _{\infty }},\quad t\in \mathbb{R}. $$
The q-hypergeometric functions are represented by
$$\begin{aligned} _{r}\phi_{s}\left ( \left . \begin{matrix} a_{1},a_{2},\ldots,a_{r} \\ \alpha_{1},\alpha_{2},\ldots,\alpha_{s} \end{matrix} \right\vert q,z \right) =&\sum_{0}^{\infty } \frac{ ( a_{1},a_{2},\ldots,a _{r};q ) _{n}}{ ( \alpha_{1},\alpha_{2},\ldots,\alpha_{s};q ) _{n}}\frac{z^{n}}{ ( q;q ) _{n}} \end{aligned}$$
and
$$\begin{aligned} {}_{m-k}\Phi_{m-1}\left ( \left . \begin{matrix} a_{1},a_{2},\ldots,a_{m-k} \\ \alpha_{1},\alpha_{2},\ldots,\alpha_{m-1} \end{matrix} \right\vert q,z \right ) =&\sum_{0}^{\infty } \frac{ ( a_{1},\ldots,a _{m-k};q ) _{n}}{ ( \alpha_{1},\ldots,\alpha_{m-1};q ) _{n}} \bigl[ ( -1 ) ^{n}q^{\binom{n}{2}} \bigr] ^{k} \\ & {} \times \frac{z^{n}}{ ( q;q ) _{n}}, \end{aligned}$$
where \(( a_{1},a_{2},\ldots,a_{p};q ) _{n}=\prod_{k=0} ^{p} ( a_{k};q ) _{n}\).

2 H-Function and related functions

The H-function, which is an extension of the hypergeometric functions \(_{p}F_{q}\), introduced by Fox [3] (see also [4, 5]), has found various applications in a huge range of problems associated with reaction, reaction diffusion, communication, engineering, fractional differential equations, integral equations, theoretical physics, and statistical distribution theory as well. The H-functions have also been recognized to play a fundamental role in fractional calculus with its applications. Fox’s H-function, admitting to a standard notation, is presented as
$$ H_{p,q}^{m,n} ( \eta ) =\frac{1}{2\pi i} \int _{P} \jmath_{p,q}^{m,n} ( w ) \eta^{w}\,dw, $$
(5)
where P is a suitable complex path, \(\eta^{w}=\exp \{ w ( \log \vert \eta \vert +i\arg \eta ) \} \), \(\jmath_{p,q}^{m,n} ( w ) =\frac{A ( w ) B ( w ) }{C ( s ) D ( w ) }\), and
$$\begin{aligned}& A ( w ) =\prod_{1}^{m}\Gamma ( b_{j}-\beta_{j}w ) ,\quad\quad B ( w ) =\prod _{1}^{n}\Gamma ( 1-a_{j}+ \alpha_{j}w ), \\& C ( w ) =\prod_{m+1}^{q}\Gamma ( 1-b_{j}-\beta_{j}w ) , \quad\quad D ( w ) =\prod _{n+1} ^{p}\Gamma ( a_{j}+ \alpha_{j}w ) , \end{aligned}$$
\(0\leq n\leq p\), \(1\leq m\leq q\), \(\{ a_{j},b_{j} \} \in \mathbb{C} \), \(\{ \alpha_{j},\beta_{j} \} \in \mathbb{R} ^{+}\). Let \(\alpha_{j}\) and \(\beta_{j}\) be positive integers and \(0\leq m\leq N\); \(0\leq n\leq M\). Then the q-analogue of Fox’s H-function is given as (see [6])
$$\begin{aligned}& H_{M,N}^{m,n}\left ( x;q\left\vert \begin{matrix} ( a_{1},\alpha_{1} ) , ( a_{2},\alpha_{2} ) ,\ldots, ( a_{\mu },\alpha_{M} ) \\ ( b_{1},\beta_{1} ) , ( b_{2},\beta_{2} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \\& \quad =\frac{1}{2\pi i} \int _{C}\frac{\prod_{j=1}^{m}G ( q^{b_{j}-\beta_{j}s} ) \prod_{j=1}^{n}G ( q^{1-a_{j}+\alpha_{j}s} ) \pi x^{s}}{\prod_{j=m+1} ^{N}G ( q^{1-b_{j}+\beta_{j}s} ) \prod_{j=n+1}^{M}G ( q^{a_{j}-\alpha_{j}s} ) G ( q^{1-s} ) \sin \pi s}\,d_{q}s, \end{aligned}$$
where G is defined in terms of the product
$$ G \bigl( q^{\alpha } \bigr) =\prod_{k=0}^{\infty } \bigl( 1-q ^{\alpha -k} \bigr) ^{-1}=\frac{1}{ ( q^{\alpha };q ) _{\infty }}. $$
(6)
The contour C is parallel to \(\operatorname{Re} ( ws ) =0\), such that all poles of \(G ( q^{b_{j}-\beta_{j}s} ) \), \(1\leq j\leq m\), are its right and those of \(G ( q^{1-a_{j}+\alpha_{j}s} ) \), \(1\leq j\leq n\), are the left of C. The above integral converges if \(\operatorname{Re} ( s\log x-\log \sin \pi s ) <0\), for huge values of \(\vert s \vert \) on C. Hence,
$$ \bigl\vert \arg ( x ) -w_{2}w_{1}^{-1}\log \vert x \vert \bigr\vert < \pi , \quad\quad \vert q \vert < 1, \quad\quad \log q=-w=-w_{1}-iw _{2}, $$
where \(w_{1}\) and \(w_{2}\) are real numbers.
Indeed, for \(\alpha_{i}=\beta_{j}=1\), for all i, j, we write the q-analogue of Meijer’s G-function as
$$ \begin{aligned}[b] &G_{M,N}^{m,n}\left ( x;q \left\vert \begin{matrix} a_{1},a_{1},\ldots,a_{M} \\ b_{1},b_{2},\ldots,b_{N} \end{matrix} \right . \right ) \\ &\quad =\frac{1}{2\pi i} \int _{C}\frac{\prod_{j=1}^{m}G ( q^{b_{j}-s} ) \prod_{j=1}^{n}G ( q^{1-a_{j}+s} ) \pi x^{2}}{\prod_{j=m+1}^{N}G ( q^{1-b_{j}+s} ) \prod_{j=n+1}^{M}G ( q^{a_{j}-s} ) G ( q^{1-s} ) \sin \pi s}\,d_{q}s, \end{aligned} $$
(7)
where \(0\leq m\leq N\); \(0\leq n\leq M\) and \(\operatorname{Re} ( s\log x-\log \sin \pi s ) <0\).
Additionally, the q-analogues of the Bessel function \(J_{v} ( x ) \) of the first kind, the Bessel function of \(Y_{v} ( x ) \), the Bessel function of the third kind \(K_{v} ( x ) \), and Struve’s function \(H_{v} ( x ) \) are, respectively, defined in terms of Fox’s \(H_{q}\)-function by [7] as follows:
$$\begin{aligned}& J_{v} ( x;q ) = \bigl\{ G ( a ) \bigr\} ^{2}H_{0,3} ^{1,0}\left ( \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} \\ ( \frac{v}{2},1 ) , ( -\frac{v}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned}$$
(8)
$$\begin{aligned}& \begin{aligned}[b] Y_{v} ( x;q ) &= \bigl\{ G ( a ) \bigr\} ^{2} \\ & \quad{} \times H_{1,4}^{2,0}\left ( \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} ( -\frac{v-1}{2},1 ) \\ ( \frac{v}{2},1 ) , ( -\frac{v}{2},1 ) ( - \frac{v-1}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned}& K_{v} ( x;q ) = ( 1-q ) H_{0,3}^{2,0}\left ( \frac{x ^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} \\ ( \frac{v}{2},1 ) , ( -\frac{v}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned}$$
(10)
$$\begin{aligned}& \begin{aligned}[b] H_{v} ( x;q ) &= \biggl( \frac{1-q}{2} \biggr) ^{1-\alpha } \\ & \quad{} \times H_{1,4}^{3,1}\left ( \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} ( \frac{1+\alpha }{2},1 ) \\ ( \frac{v}{2},1 ) , ( -\frac{v}{2},1 ) ( \frac{v+ \alpha }{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned} \end{aligned}$$
(11)
In [8] (see also [9]), some q-analogues of the natural exponential functions, sine functions, cosine functions, hyperbolic sine functions, and hyperbolic cosine functions are, respectively, given in terms of Fox′s H-function as follows:
$$\begin{aligned}& e_{q} ( -x ) =G ( q ) H_{0,2}^{1,0}\left ( x ( 1-q ) ;q \left\vert \textstyle\begin{array}{l} \\ ( 0,1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned}$$
(12)
$$\begin{aligned}& \begin{aligned}[b] \sin_{q} ( x ) &= \sqrt{\pi } ( 1-q ) ^{- \frac{1}{2}} \bigl\{ G ( q ) \bigr\} ^{2} \\ & \quad{} \times H_{0,3}^{1,0}\left ( \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} ( \frac{1}{2},1 ) ( 0,1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned}& \begin{aligned}[b] \cos_{q} ( x ) &= \sqrt{\pi } ( 1-q ) ^{- \frac{1}{2}} \bigl\{ G ( q ) \bigr\} ^{2} \\ & \quad{} \times H_{0,3}^{1,0}\left ( \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} \\ ( 0,1 ) ( \frac{1}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \end{aligned}$$
(14)
$$\begin{aligned}& \begin{aligned}[b] \sinh_{q} ( x ) &= \frac{\sqrt{\pi }}{i} ( 1-q ) ^{-\frac{1}{2}} \bigl\{ G ( q ) \bigr\} ^{2} \\ & \quad{} \times H_{0,3}^{1,0}\left ( - \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} \\ ( \frac{1}{2},1 ) ( 0,1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned}& \begin{aligned}[b] \cosh_{q} ( x ) &= \sqrt{\pi } ( 1-q ) ^{- \frac{1}{2}} \bigl\{ G ( q ) \bigr\} ^{2} \\ & \quad{} \times H_{0,3}^{1,0}\left ( - \frac{x^{2} ( 1-q ) ^{2}}{4};q \left\vert \textstyle\begin{array}{l} \\ ( 0,1 ) ( \frac{1}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned} \end{aligned}$$
(16)

On the other hand, some impressive integral transforms also have the corresponding q-analogues in the concept of q-calculus; they include the q-Laplace transforms [10], the q-Sumudu transforms [9, 11, 12, 13], the q-Wavelet transform [14], the q-Mellin transform [15], q-\(E_{2,1}\)-transform [16], q-Mangontarum transforms [17, 18], q-natural transforms [19], and so on. Recently, a number of authors have studied various image formulas for these q-integral transforms, associated with a variety of special functions. In this sequel, we aim to investigate the q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox’s \(H_{q}\)-function.

3 q-Laplace-type transforms for \(H_{q}\)-function

A Laplace-type integral was introduced in [20, 21]. The q-analogues of the Laplace-type integral of the first kind were defined later by [22] as follows:
$$\begin{aligned} _{q}L_{2} \bigl( f ( \xi ) ;y \bigr) =& \frac{1}{1-q^{2}} \int _{0}^{y^{-1}}\xi E_{q^{2}} \bigl( q^{2}y^{2}\xi^{2} \bigr) f ( \xi ) \,d\xi \\ =& \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y ^{2}}\sum_{i=0}^{\infty } \frac{q^{2i}}{ ( q^{2};q^{2} ) _{i}}f \bigl( q^{i}y^{-1} \bigr) , \end{aligned}$$
(17)
whereas the q-analogues of the Laplace-type integral of the second kind were defined by
$$\begin{aligned} _{q}\ell_{2} \bigl( f ( \xi ) ;y \bigr) =& \frac{1}{1-q^{2}} \int _{0}^{\infty }\xi e_{q^{2}} \bigl( y^{2}\xi^{2} \bigr) \,d _{q}\xi \\ =& \frac{1}{ [ 2 ] _{q} ( -y^{2};q^{2} ) _{\infty }}\sum_{i\in \mathbb{Z} }q^{2i}f \bigl( q^{i} \bigr) \bigl( -y^{2};q ^{2} \bigr) _{i}. \end{aligned}$$
(18)
For the sake of convenience, we establish some formulas for the \(_{q}L_{2}\) operator. A similar argument can give certain corresponding results for the operator \(_{q}\ell_{2}\).

Theorem 1

Letβbe a positive real number. Then
$$ _{q}L_{2} \bigl( \xi^{2\beta -2} \bigr) ( y ) = \frac{ ( q ^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y^{2} ( q^{ \beta };q^{2} ) _{\infty }} . $$

Proof

By using (17), we have
$$\begin{aligned} _{q}L_{2} \bigl( \xi^{2\beta -2};y \bigr) =& \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y^{2\beta }}\sum_{i=0}^{\infty } \frac{q^{2i}}{ ( q^{2};q^{2} ) _{i}} \bigl( q ^{i}y^{-1} \bigr) ^{2\beta -2} \\ =&\frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y ^{2\beta }}\sum_{i=0}^{\infty } \frac{q^{2\beta i}y^{2\beta -2}}{ ( q^{2};q^{2} ) _{i}}. \end{aligned}$$
That is,
$$\begin{aligned} _{q}L_{2} \bigl( \xi^{2\beta -2};y \bigr) =& \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y^{2}}\sum_{i=0} ^{\infty } \frac{q^{2\beta i}}{ ( q^{2};q^{2} ) _{i}} . \end{aligned}$$
(19)
By the fact that
$$ e_{q} ( z ) =\sum_{i=0}^{\infty } \frac{z^{i}}{ ( q;q ) _{i}}, $$
we have
$$\begin{aligned} _{q}L_{2} \bigl( \xi^{2\beta -2};y \bigr) =& \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y^{2}}e_{q^{2}} \bigl( q^{\beta } \bigr) \\ =& \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q}y ^{2}}\frac{1}{ ( q^{2\beta };q^{2} ) _{\infty }}. \end{aligned}$$
This completes the establishment of the belief. □

Theorem 2

Letλbe a complex number. Then
$$\begin{aligned}& _{q}L_{2}\left ( x^{2\lambda }H_{M,N}^{m,n} \left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \right) ( y ) \\& \quad =\frac{ ( q^{2};q^{2} ) _{\infty }}{y ^{2\lambda +2} [ 2 ] _{q} }H_{M+1,N}^{m,n+1}\left ( \frac{ \gamma }{y^{2k}},q^{2} \left\vert \textstyle\begin{array}{l} ( -\lambda ,k ) , ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{array}\displaystyle \right . \right ) , \end{aligned}$$
where\(0\leq n\leq m\)and\(0\leq m\leq N\)andλis an arbitrary complex number.

Proof

Let λ be a complex number. Then by (17) we obtain
$$\begin{aligned}& _{q}L_{2}\left ( x^{2\lambda }H_{M,N}^{m,n} \left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \right) ( y ) \\& \quad =\frac{1}{2\pi i} \int _{c}\frac{\prod _{j=1}^{m}G ( q^{2b_{j}-2 \beta_{j}z} ) \prod _{j=1}^{n}G ( q^{2-2a_{j}+2\alpha_{j}z} ) \pi \gamma^{z}}{\prod _{j=m+1}^{N}G ( q^{2-2b_{j}+2\beta_{j}z} ) \prod _{j=n+1}^{M}G ( q^{2a_{j}-2\alpha_{j}z} ) G ( q^{2-2z} ) \sin \pi z} \\& \quad\quad{} \times_{q}L_{2} \bigl( x^{2\lambda +2 k z} \bigr) (y) \,d_{q}z. \end{aligned}$$
(20)
Let \(\beta =\lambda +k z+1\), then by Theorem 1 we have
$$ _{q}L_{2} \bigl( x^{2 ( \lambda +kz ) } \bigr) ( y ) = _{q}L_{2} \bigl( x^{2B-2} \bigr) ( y ) = \frac{ ( q^{2};q ^{2} ) _{\infty }}{ [ 2 ] _{q} y^{2} ( q^{2 ( \lambda +zk+1 ) };q^{2} ) _{\infty }} . $$
(21)
By invoking (21) in (20), we get
$$\begin{aligned}& _{q}L_{2}\left ( x^{2\lambda }H_{M,N}^{m,n} \left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \right) ( y ) \\& \quad =\frac{1}{2\pi i} \int _{c}\frac{\prod _{j=1}^{m}G ( q^{2b_{j}-2 \beta_{j}z} ) \prod _{j=1}^{n}G ( q^{2-2a_{j}+2\alpha_{j}z} ) \pi \gamma^{z}}{\prod _{j=m+1}^{N}G ( q^{2-2b_{j}+2\beta_{j}z} ) \prod _{j=n+1}^{M}G ( q^{2a_{j}-2\alpha_{j}z} ) G ( q^{2-2z} ) \sin \pi z} \\& \quad\quad{} \times \frac{ ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q} y^{2} ( q^{2 ( \lambda +zk+1 ) };q^{2} ) _{\infty }} \,d_{q}z. \end{aligned}$$
(22)
By inserting the identity
$$ G \bigl( q^{2\lambda +2kz+2} \bigr) =\frac{1}{ ( q^{2\lambda +2kz+2};q ^{2} ) _{\infty }} $$
in (22) yields
$$\begin{aligned}& _{q}L_{2}\left ( x^{2\lambda }H_{M,N}^{m,n} \left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \right) ( y ) \\& \quad =\frac{ ( q^{2};q^{2} ) _{\infty }}{2\pi iy^{2 \lambda +2} [ 2 ] _{q} } \int _{c}\frac{\prod _{j=1}^{m}G ( q^{2b_{j}-2\beta_{j}z} ) \prod _{j=1}^{n}G ( q^{2-2a_{j}+2 \alpha_{j}z} ) }{\prod _{j=m+1}^{N}G ( q^{2-2b_{j}+2\beta_{j}z} ) \prod _{j=n+1}^{M}G ( q^{2a_{j}-2 \alpha_{j}z} ) } \\& \quad\quad{}\times \frac{G ( q^{1+\lambda +kz} ) }{G ( q^{2 ( 1-z ) } ) \sin \pi z}\pi \biggl( \frac{\gamma }{y^{2}k} \biggr) ^{z}\,d_{q}z. \end{aligned}$$
Now, on account of the definition of \(H_{q}\)-function, we may establish that
$$\begin{aligned}& _{q}L_{2}\left ( x^{2\lambda }H_{M,N}^{m,n} \left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( a_{1},\alpha_{1} ) ,\ldots, ( a_{M},\alpha_{M} ) \\ ( b_{1},\beta_{1} ) ,\ldots, ( b_{N},\beta_{N} ) \end{matrix} \right . \right ) \right) ( y ) \\& \quad =\frac{ ( q^{2};q^{2} ) _{\infty }}{y^{2\lambda +2} [ 2 ] _{q} }H_{N,M+1}^{n+1,m}\left ( \gamma x^{2k};q^{2} \left\vert \begin{matrix} ( 1-b_{1},\beta_{1} ) ,\ldots, ( 1-b_{N},\beta_{N} ) \\ ( 1+\lambda ,k ) , ( 1-a,\alpha_{1} ) ,\ldots, ( 1-a _{M},\alpha_{M} ) \end{matrix} \right . \right) , \end{aligned}$$
provided \(k<0\).

The proof is completed. □

4 Applications to trigonometric and hyperbolic functions

In this part, we shall give certain natural relevance to the leading results.

Theorem 3

Let\(e_{q}\)be defined in terms of (12). Then
$$ _{q}L_{2} \bigl( e_{q^{2}} ( -x ) \bigr) ( y ) = \frac{G ( q^{2} ) ( q^{2};q^{2} ) _{\infty }}{ [ 2 ] _{q} y^{2}}H_{1,2}^{1,1}\left ( \frac{1-q^{2}}{y^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( 0,1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . $$

Proof

By setting \(\lambda =0\), \(\gamma =1-q^{2}\), and \(k=1\), Theorem 3 immediately follows from Theorem 2. □

The demonstration of this theorem is finished.

Theorem 4

Let\(\sin_{q}\)be defined in terms of (13). Then we have
$$\begin{aligned} _{q}L_{2} \bigl( \sin_{q^{2}} ( x ) \bigr) ( y ) &=\frac{\sqrt{ \pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q^{2} ) \} ^{2}}{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{1,3}^{1,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( \frac{1}{2},1 ) ( 0,1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned}$$

Proof

The proof of this theorem indeed follows from substituting the values \(\lambda =0\), \(k=1\), and \(\gamma = \frac{ ( 1-q^{2} ) ^{2}}{4} \) and from multiplying by \(\sqrt{\pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q ^{2} ) \} ^{2}\).

Hence, the proof is completed. □

Theorem 5

Let\(\cos_{q}\)be defined in terms of (14). Then
$$\begin{aligned} _{q}L_{2} \bigl( \cos_{q^{2}} ( x ) \bigr) ( y ) &=\frac{\sqrt{ \pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q^{2} ) \} ^{2}}{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{1,3}^{1,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y ^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( 0,1 ) , ( \frac{1}{2},1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned}$$

Proof

Proof follows from Theorem 2 for \(\lambda =0\), \(k=1\), \(\gamma =\frac{ ( 1-q^{2} ) ^{2}}{4}\).

The proof is completed. □

Theorem 6

Let\(\sinh_{q}\)be defined in terms of (15). Then
$$\begin{aligned} _{q}L_{2} \bigl( \sinh_{q^{2}} ( x ) \bigr) ( y ) &=\frac{\sqrt{ \pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q^{2} ) \} ^{2}}{i [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{1,3}^{1,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q ^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( \frac{1}{2},1 ) ( 0,1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned}$$

Proof

By using the special case, \(\lambda =0\), \(k=1\), \(\gamma =\frac{ ( 1-q ^{2} ) ^{2}}{4}\).

The proof is completed. □

Theorem 7

Let\(\cosh_{q}\)be defined in terms of (16). Then
$$\begin{aligned} _{q}L_{2} \bigl( \cosh_{q^{2}} ( x ) \bigr) ( y ) &=\frac{\sqrt{ \pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q^{2} ) \} ^{2}}{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H _{1,3}^{1,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( 0,1 ) , ( \frac{1}{2},1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned}$$

Proof

The validation of this theorem is identical to that of the previous theorem. □

Theorem 8

Let the Bessel function be defined in terms of (8). Then
$$\begin{aligned} _{q}L_{2} \bigl( J_{v} \bigl( x;q^{2} \bigr) \bigr) ( y ) &=\frac{ \{ G ( q^{2} ) \} ^{2}}{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{1,3}^{1,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q ^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( \frac{v}{2},1 ) , ( \frac{-v}{2},1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned}$$

Proof

By setting \(\lambda =0\), \(k=1\), \(\gamma =\frac{1-q^{2}}{4}\) and multiplying by \(\{ G ( q^{2} ) \} ^{2}\), the result follows. □

Theorem 9

Let theq-Bessel function of the second kind be defined in terms of (9)(11). Then
$$\begin{aligned}& \begin{aligned} _{q}L_{2} \bigl( Y_{v} \bigl( x;q^{2} \bigr) \bigr) ( y ) &= \frac{ \{ G ( q^{2} ) \} ^{2}}{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{2,4}^{2,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) , ( \frac{-v}{2},1 ) \\ ( \frac{v}{2},1 ) , ( \frac{-v}{2},1 ) , ( \frac{-v-1}{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \\& \begin{aligned} _{q}L_{2} \bigl( K_{v} \bigl( x;q^{2} \bigr) \bigr) ( y ) & =\frac{ ( 1-q^{2} ) }{ [ 2 ] _{q} y^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{}\times H_{1,3}^{2,1}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q^{2} \left\vert \textstyle\begin{array}{l} ( 0,1 ) \\ ( \frac{v}{2},1 ) , ( \frac{-v}{2},1 ) , ( 1,1 ) \end{array}\displaystyle \right . \right ) , \end{aligned} \\& \begin{aligned} _{q}L_{2} \bigl( H_{v} \bigl( x;q^{2} \bigr) \bigr) ( y ) & =\frac{ ( 1-q^{2} ) ^{1-\alpha }}{2^{1-\alpha } [ 2 ] _{q} y ^{4}} \bigl( q^{2};q^{2} \bigr) _{\infty } \\ &\quad{} \times H_{2,4}^{3,2}\left ( \frac{ ( 1-q^{2} ) ^{2}}{4y^{2}};q ^{2}\left\vert \textstyle\begin{array}{l} ( 0,1 ) , ( \frac{1-\alpha }{2},1 ) \\ ( \frac{v}{2},1 ) , ( \frac{-v}{2},1 ) , ( \frac{1+ \alpha }{2},1 ) ( 1,1 ) \end{array}\displaystyle \right . \right ) . \end{aligned} \end{aligned}$$

Proof

Proof of this theorem follows from (9)–(11) and the technique quite similar to that of Theorems 38. We omit the details. □

Notes

Acknowledgements

The authors are thankful to the referee for his/her valuable remarks and comments for the improvement of the paper.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Kac, V.G., Cheung, P.: Quantum Calculus: Universitext. Springer, New York (2002) CrossRefMATHGoogle Scholar
  2. 2.
    Gasper, G., Rahman, M.: Generalized Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  3. 3.
    Fox, C.: The G and H-functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. 98, 395–429 (1961) MATHMathSciNetGoogle Scholar
  4. 4.
    Mathai, A.M., Saxena, R.K.: Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer, Berlin (1973) CrossRefMATHGoogle Scholar
  5. 5.
    Mathai, A.M., Saxena, R.K.: The H-Function with Application in Statistics and Other Disciplines. Wiley, New York (1978) MATHGoogle Scholar
  6. 6.
    Saxena, R.K., Modi, G.C., Kalla, S.L.: A basic analogue of Fox’s H-function. Rev. Téc. Fac. Ing., Univ. Zulia 6, 139–143 (1983) MATHMathSciNetGoogle Scholar
  7. 7.
    Saxena, R.K., Kumar, R.: Recurrence relations for the basic analogue of the H-function. J. Nat. Acad. Math. 8, 48–54 (1990) MATHMathSciNetGoogle Scholar
  8. 8.
    Yadav, R.K., Purohit, S.D., Kalla, S.L.: On generalized Weyl fractional q-integral operator involving generalized basic hypergeometric functions. Fract. Calc. Appl. Anal. 11(2), 129–142 (2008) MATHMathSciNetGoogle Scholar
  9. 9.
    Albayrak, D., Purohit, S.D., Ucar, F.: On q-Sumudu transforms of certain q-polynomials. Filomat 27(2), 413–429 (2013) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Abdi, W.H.: On q-Laplace transforms. Proc. Natl. Acad. Sci., India 29, 389–408 (1961) MATHGoogle Scholar
  11. 11.
    Albayrak, D., Purohit, S.D., Ucar, F.: On q-analogues of Sumudu transform. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 21(1), 239–260 (2013) MATHMathSciNetGoogle Scholar
  12. 12.
    Albayrak, D., Purohit, S.D., Ucar, F.: Certain inversion and representation formulas for q-Sumudu transforms. Hacet. J. Math. Stat. 43(5), 699–713 (2014) MATHMathSciNetGoogle Scholar
  13. 13.
    Purohit, S.D., Ucar, F.: An application of q-Sumudu transform for fractional q-kinetic equation. Turk. J. Math. 42(1), 726–734 (2018) Google Scholar
  14. 14.
    Fitouhi, A., Bettaibi, N.: Wavelet transforms in quantum calculus. J. Nonlinear Math. Phys. 13(3), 492–506 (2006) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Fitouhi, A., Bettaibi, N.: Applications of the Mellin transform in quantum calculus. J. Math. Anal. Appl. 328, 518–534 (2007) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Salem, A., Ucar, F.: The q-analogue of the \(E_{2,1}\)-transform and its applications. Turk. J. Math. 40(1), 98–107 (2016) CrossRefGoogle Scholar
  17. 17.
    Mangontarum, M.M.: On a q-analogue of the Elzaki transform called Mangontarum q-transform. Discrete Dyn. Nat. Soc. 2014, Article ID 825618 (2014) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Al-Omari, S.K.Q.: On q-analogues of the Mangontarum transform for certain q-Bessel functions and some application. J. King Saud Univ., Sci. 28(4), 375–379 (2016) CrossRefGoogle Scholar
  19. 19.
    Al-Omari, S.K.Q.: On q-analogues of the Natural transform of certain q-Bessel functions and some application. Filomat 31(9), 2587–2598 (2017) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Yürekli, O.: Theorems on \(L_{2}\)-transforms and its applications. Complex Var. Elliptic Equ. 38, 95–107 (1999) MATHGoogle Scholar
  21. 21.
    Yürekli, O.: New identities involving the Laplace and the \(L_{2}\)-transforms and their applications. Appl. Math. Comput. 99, 141–151 (1999) MATHGoogle Scholar
  22. 22.
    Ucar, F., Albayrak, D.: On q-Laplace type integral operators and their applications. J. Differ. Equ. Appl. 18(6), 1001–1014 (2012) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Shrideh K. Q. Al-Omari
    • 1
  • Dumitru Baleanu
    • 2
    • 3
  • Sunil D. Purohit
    • 4
  1. 1.Department of Basic Sciences, Faculty of Engineering TechnologyAl-Balqa Applied UniversityAmmanJordan
  2. 2.Department of MathematicsCankaya UniversityAnkaraTurkey
  3. 3.Institute of Space SciencesMagurele-BucharestRomania
  4. 4.Department of HEAS (Mathematics)Rajasthan Technical UniversityKotaIndia

Personalised recommendations