Some results for Laplace-type integral operator in quantum calculus

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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.

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 , see also ):

$$\int _{0}^{a}f ( x ) \,d_{q}x= ( 1-q ) a\sum _{0}^{\infty }f \bigl( aq^{k} \bigr) q^{k}$$
(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 ):

$$\int _{0}^{\frac{\infty }{A}}f ( x ) \,d_{q}x= ( 1-q ) \sum _{n\in \mathbb{Z} }\frac{q^{k}}{A}f \biggl( \frac{q^{k}}{A} \biggr) , \quad A\in \mathbb{C}.$$

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. )

$$\Gamma_{q} ( x ) =\frac{ ( q;q ) _{\infty }}{ ( q^{x};q ) _{\infty }} ( 1-q ) ^{1-x}\quad \text{and}\quad ( a;q ) _{t}=\frac{ ( a;q ) _{\infty }}{ ( aq^{t};q ) _{\infty }},\quad t\in \mathbb{R}.$$

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 $$( a_{1},a_{2},\ldots,a_{p};q ) _{n}=\prod_{k=0} ^{p} ( a_{k};q ) _{n}$$.

H-Function and related functions

The H-function, which is an extension of the hypergeometric functions $$_{p}F_{q}$$, introduced by Fox  (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 )

\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  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  (see also ), 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 , the q-Sumudu transforms [9, 1113], the q-Wavelet transform , the q-Mellin transform , q-$$E_{2,1}$$-transform , q-Mangontarum transforms [17, 18], q-natural transforms , 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.

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  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

Let β be 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

Let λ be 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 $$0\leq m\leq N$$ and λ is 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. □

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 the q-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. □

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Acknowledgements

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

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

Correspondence to Sunil D. Purohit.

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