Theory and Modern Applications

# A fractional q-integral operator associated with a certain class of q-Bessel functions and q-generating series

## Abstract

This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various types of q-Bessel functions and some power series of special type. It has been obtained for basic q-generating series, q-exponential and q-trigonometric functions as well. Various results and corollaries are provided as an application to this theory.

## Introduction

The theory of q-calculus is an old subject centered on the idea of deriving q-analogous results without using limits. Jackson was the first to develop the q-calculus theory in systematic way . He defined the concept of the q-integral and the concept of the q-difference operator in a generic manner. In excellence, the theory of q-calculus allows to deal with sets of non-differentiable functions, different classes of orthogonal polynomials, integral operators, and various classes of special functions including q-hypergeometric functions, q-Bessel functions, q-gamma and q-beta functions, and many others, to mention but a few. It connects mathematics and physics and plays a significant role in various fields of physical sciences such as cosmic strings , conformal quantum mechanics , and nuclear physics of high energy . It, further, applies to topics in number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, quantum theory, mechanics, and the theory of relativity.

The q-integrals from 0 to ξ and from 0 to ∞ are, resp., defined by Jackson as 

$$\int _{0}^{\xi }f ( t ) \,d_{q}t=\xi ( 1-q ) \sum_{j=0}^{\infty }q^{j}f \bigl( \xi q^{j} \bigr)$$
(1)

and

$$\int _{0}^{\infty /A}f ( t ) \,d_{q}t= ( 1-q ) \sum_{j\in \mathbb{Z} }\frac{q^{j}}{A}f \biggl( \frac{q^{j}}{A} \biggr) .$$
(2)

The q-analogue of the Bessel function

$$J_{\mu } ( \xi ) =\sum_{j=0}^{\infty } \frac{ ( -1 ) ^{j} ( \frac{\xi }{2} ) ^{\mu +2j}}{j!\Gamma ( \mu +j+1 ) }$$
(3)

of the first type, which was studied later by Hahn  and Ismail , is defined by  as

$$J_{\mu }^{ ( 1 ) } ( \xi ;q ) = \biggl( \frac{\xi }{2} \biggr) ^{\mu }\sum_{j=0}^{\infty } \frac{ ( \frac{-\xi }{4}^{2} ) ^{j}}{ ( q,q ) _{\mu +j} ( q;q ) _{j}}, \quad \vert \xi \vert < 2.$$
(4)

Jackson defines the q-analogue of the Bessel function of the second type as 

$$J_{\mu }^{ ( 2 ) } ( \xi ;q ) = \biggl( \frac{\xi }{2} \biggr) ^{\mu }\sum_{j=0}^{\infty } \frac{q^{j ( j+\mu ) } ( \frac{-\xi }{4}^{2} ) ^{j}}{ ( q;q ) _{\mu +j} ( q;q ) _{j}}, \quad \xi \in \mathbb{C} .$$
(5)

Hahn  and Exton  introduced the third type q-Bessel function (called Hahn–Exton q-Bessel function) as

$$J_{\mu }^{ ( 3 ) } ( \xi ;q ) =\xi ^{\mu }\sum _{j=0}^{ \infty } \frac{ ( -1 ) ^{j}q^{\frac{j ( j-1 ) }{2}} ( q\xi ^{2} ) ^{j}}{ ( q;q ) _{\mu +j} ( q;q ) _{j}}, \quad \xi \in \mathbb{C} .$$
(6)

The q-shifted factorials are defined, in literature, by fixing $$\xi \in \mathbb{C}$$ as

$$( \xi ;q ) _{0}=1; \quad\quad ( \xi ;q ) _{n}= \prod _{j=0}^{n-1} \bigl( 1-\xi q^{k} \bigr) , \quad n=1,2,\ldots; \quad\quad ( \xi ;q ) _{\infty }= \underset{n \rightarrow \infty }{\lim } ( \xi ;q ) _{n}.$$
(7)

This indeed gives

$$( \xi ;q ) _{x}= \frac{ ( \xi ;q ) _{\infty }}{ ( \xi q^{x};q ) _{\infty }}, \quad x\in \mathbb{R} .$$
(8)

For $$\xi \in \mathbb{C}$$, we mean

$$[ \xi ] _{q}=\frac{1-q^{\xi }}{1-q}.$$

Hence, for $$n\in \mathbb{N}$$, we obtain

$$\bigl( [ n ] _{q} \bigr) != \frac{ ( q;q ) _{n}}{ ( 1-q ) ^{n}}.$$

Due to [10, (1.5), (1.6)], we, resp., write

$$\begin{bmatrix} n \\ k\end{bmatrix} _{q}= \frac{ [ n ] _{q}!}{ [ k ] _{q}! [ n-k ] _{q}!}= \frac{ ( q;q ) _{n}}{ ( q;q ) _{k} ( q;q ) _{n-k}}$$
(9)

and

$$\begin{bmatrix} \alpha \\ k\end{bmatrix} _{q}= \frac{ ( q^{-\alpha };q ) _{k}}{ ( q;q ) _{k}} ( -1 ) ^{k}q^{\alpha k- \binom{ k}{ 2} }= \frac{\Gamma _{q} ( \alpha +1 ) }{\Gamma _{q} ( k+1 ) \Gamma _{q} ( \alpha -k ) }.$$
(10)

The q-analogue of the exponential function of the second type is given by

$$e_{q} ( \xi ) =\sum_{j=0}^{\infty } \frac{\xi ^{j}}{ ( q;q ) _{j}}=\frac{1}{ ( \xi ;q ) _{\infty }}, \quad \vert \xi \vert < 1,$$
(11)

whereas the q-analogue of the exponential function of the first type is given by

$$E_{q} ( \xi ) =\sum_{j=0}^{\infty } \frac{ ( -1 ) ^{j}q^{j\frac{ ( j-1 ) }{2}}\xi ^{j}}{ ( q;q ) _{j}}= ( \xi ;q ) _{\infty }, \quad \xi \in \mathbb{C} .$$

Consequently, the following formula holds:

$$\bigl( q^{\xi +m};q \bigr) _{\infty }= \frac{ ( q^{\xi };q ) _{\infty }}{ ( q^{\xi };q ) _{m}},\quad m\in \mathbb{N} .$$
(12)

For real arguments t, the q-analogues of the gamma function are given by 

$$\Gamma _{q} ( t ) = \int _{0}^{\frac{1}{1-q}}x^{t-1}E_{q} ( -qx ) \,d_{q}x\quad \text{and}\quad \hat{\Gamma }_{q} ( t ) = \int _{0}^{\infty }x^{t-1}e_{q} ( -x ) \,d_{q}x.$$
(13)

Henceforth, for $$t\in \mathbb{R}$$ and $$n\in \mathbb{N}$$, the following auxiliary results hold:

$$\Gamma _{q} ( t+1 ) = [ t ] _{q}\Gamma _{q} ( t ) , \quad\quad \Gamma _{q} ( n+1 ) = [ n ] _{q}!\quad \text{and}\quad \Gamma _{q} ( t+1 ) =\frac{1-q^{t}}{1-q}\Gamma _{q} ( t ) .$$
(14)

The theory of fractional calculus was born in early 1695 due to a very deep question raised in a letter of L’Hospital to Leibniz . During a long period of time (300 years), the fractional calculus has kept the attention of top level mathematicians. It has become a very useful tool for tackling dynamics of complex systems from various branches of science and engineering. The fractional q-calculus is the q-extension of the ordinary fractional calculus. Integral operators have attained their popularity due to their wide range of applications in various fields of science and engineering  and . In [35, 36] Al-Salam and Agarwal studied certain q-fractional integrals and derivatives. Recently, perhaps due to explosion in research within the fractional calculus setting, new developments in the theory of fractional q-difference calculus, specifically, the q-analogues of the integral and the differential fractional operator properties were made, see, e.g., . In [36, p. 966], Al-Salam defines a fractional q-integral operator in the form of the basic integral

$$K_{q}^{\eta }f ( x ) = \frac{q^{-\eta }x^{\eta }}{\Gamma _{q} ( \alpha ) } \int _{x}^{\infty } ( y-x ) _{ \alpha -1}y^{-\eta -\alpha }f \bigl( yq^{1-\alpha } \bigr) \,d ( y;q ) ,$$
(15)

provided $$\alpha \neq 0,-1,-2,\ldots$$ . With the aid of series definition (1), the above equation can be expressed as

$$K_{q}^{\eta }f ( x ) = ( 1-q ) ^{\alpha }\sum _{k=0}^{\alpha } ( -1 ) ^{k}q^{k ( \eta + \alpha ) +\frac{1}{2}k ( k+1 ) } \begin{bmatrix} -\alpha \\ k\end{bmatrix} f \bigl( xq^{-\alpha -k} \bigr) .$$
(16)

Consequently, by applying (9), (2) can be expressed as

$$K_{q}^{\eta ,\alpha }f ( x ) = ( 1-q ) ^{ \alpha }\sum _{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta + \alpha ) +\frac{1}{2}k ( k+1 ) } \biggl( \frac{ ( q;q ) _{-\alpha }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \biggr) f \bigl( xq^{-\alpha -k} \bigr) .$$

Therefore, it follows that

$$K_{q}^{\eta ,\alpha }f ( x ) = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}f \bigl( xq^{- \alpha -k} \bigr) .$$
(17)

In what follows, we discuss the Al-Salam fractional q-integral $$( 15 )$$ on some special functions. We apply it to various types of q-Bessel functions and some power series of special type. In Sect. 1, we already recalled some definitions and notations from the fractional q-calculus theory. In Sect. 2, we apply the Al-Salam fractional q-integral to a finite product of q-Bessel functions. In Sect. 3, we apply the Al-Salam fractional q-integral to a power series. We also include some new applications. In Sect. 4, we apply the Al-Salam q-integral operator to some q-generating series.

## Main results

### Theorem 1

Let $$\{ J_{2\mu _{1}}^{ ( 1 ) } ( 2\sqrt{\delta _{1}t};q ) ,\ldots,J_{2\mu _{n}}^{ ( 1 ) } ( 2\sqrt{ \delta _{n}t};q ) \}$$ be a set of first kind q-Bessel functions and

$$f ( t ) =t^{\Delta -1}\prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 1 ) } ( 2\sqrt{\delta _{j}t};q ) .$$
(18)

Then, for some $$B=q^{-\alpha ( \Delta -1 ) } \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}$$, we have

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&B\prod_{j=1}^{n} \bigl( \delta _{j}xq^{-\alpha } \bigr) ^{\mu _{j}}\sum _{m=0}^{ \infty }\delta _{j}x^{m}q^{-\alpha m} \frac{ ( q^{2\mu _{j}+m+1};q ) _{\infty }}{\Gamma _{q} ( m+1 ) } \\ & {}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( -m+\eta +\alpha +\frac{1}{2}k+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}

### Proof

By employing (18), the fractional q-integral (17) reveals

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{k=0}^{\infty } ( -1 ) ^{k}\frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}f \bigl( xq^{- \alpha -k} \bigr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{k=0}^{\infty } \frac{ ( -1 ) ^{k}q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \bigl( xq^{{-\alpha -k}} \bigr) ^{ \Delta -1} \\ &{}\times \prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 1 ) } \bigl( 2 \sqrt{\delta _{j}xq^{-\alpha -k}};q \bigr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}\sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) - ( \alpha +k ) ( \Delta -1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \\ &{}\times \prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 1 ) } \bigl( 2 \sqrt{\delta _{j}xq^{-\alpha -k}};q \bigr) . \end{aligned}

By taking into account the definition of the Bessel function $$J_{v}^{ ( 1 ) }$$ given in (4), jointly with simple computations, the above equation reduces to yield

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&\frac{ ( q;q ) _{-\alpha }x^{\Delta -1}}{ ( 1-q ) ^{-\alpha }} \sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) - ( \alpha +k ) ( \Delta -1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \prod_{j=1}^{n} \bigl( a_{j}xq^{-\alpha -k} \bigr) ^{\mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } \frac{ ( \delta _{j}xq^{-\alpha -k} ) ^{m}}{ ( q,q ) _{2\mu _{j}+m} ( q,q ) _{m}} \\ =&q^{-\alpha ( \Delta -1 ) } \frac{ ( q;q ) _{-\alpha }x^{\Delta -1}}{ ( 1-q ) ^{-\alpha }} \prod_{j=1}^{n} \bigl( \delta _{j}xq^{-\alpha } \bigr) ^{ \mu _{j}}\sum _{m=0}^{\infty } \frac{ ( \delta _{j}xq^{-\alpha } ) ^{m}q^{-km}}{ ( q;q ) _{2\mu _{j}+m} ( q;q ) _{m}} \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{-km+k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) -k ( \Delta -1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}. \end{aligned}

Hence, by the fact [40, Equ. (8)]

$$( \zeta ;q ) _{x}= \frac{ ( \zeta ;q ) _{\infty }}{ ( \zeta q^{x};q ) _{\infty }},$$
(19)

we obtain

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&q^{-\alpha ( \Delta -1 ) } \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1} \prod_{j=1}^{n} \delta _{j}^{\mu _{j}} \mu _{j}xq^{\alpha \mu _{j}-\alpha }q^{\mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } \frac{\delta _{j}^{m}x^{m}q^{-\alpha m} ( q^{2\mu _{j}+m+1};q ) _{\infty }}{ ( q;q ) _{m}} \sum _{k=0}^{\infty } \frac{ ( -1 ) ^{k}q^{k ( -m+\eta +\alpha +\frac{1}{2}k+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}
(20)

This completes the proof of the theorem. □

Now the identity

$$( q;q ) _{\alpha }=\Gamma _{q} ( \alpha +1 )$$
(21)

leads to the following useful remark.

### Remark 2

Let $$\{ J_{2\mu _{1}}^{ ( 1 ) } ( 2\sqrt{\delta _{1}t};q ) ,\ldots,J_{2\mu _{n}}^{ ( 1 ) } ( 2\sqrt{ \delta _{n}t};q ) \}$$ be a set of first kind q-Bessel functions and $$f ( t ) =t^{\Delta -1}\prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 1 ) } ( 2\sqrt{\delta _{j}t};q )$$. Then, for some $$B=q^{-\alpha ( \Delta -1 ) } \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}$$, we have

\begin{aligned} K_{q}^{\eta ,\alpha }f(x) =&B\prod_{j=1}^{n} \bigl( \delta _{j}xq^{- \alpha } \bigr) ^{\mu _{j}}\sum _{m=0}^{\infty }\delta _{j}x^{m}q^{- \alpha m} \frac{ ( q^{2\mu _{j}+m+1};q ) _{\infty }}{\Gamma _{q} ( m+1 ) } \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( -m+\eta +\alpha +\frac{1}{2}k+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}

### Proof

Indeed, from Theorem 1 and (21), we have

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&\frac{q^{-\alpha ( \Delta -1 ) }x^{\Delta -1}}{ ( q;q ) _{\infty } ( 1-q ) ^{-\alpha }}\Gamma _{q} ( 1-\alpha ) \prod_{j=1}^{n} \bigl( \delta _{j}xq^{-\alpha \mu _{j}-\alpha } \bigr) ^{\mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } \frac{ ( \delta _{j}xq^{-\alpha } ) ^{m} ( q^{\mu _{j}+m+1};q ) _{\infty }}{\Gamma _{q} ( m+1 ) } \sum _{k=0}^{\infty } \frac{ ( -1 ) ^{k}q^{k ( -m+\eta +\alpha +\frac{1}{2}k+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( 1-\alpha -k ) } \\ =&B\prod_{j=1}^{n} \bigl( \delta _{j}xq^{-\alpha } \bigr) ^{ \mu _{j}}\sum _{m=0}^{\infty }\delta _{j}x^{m}q^{-\alpha m} \frac{ ( q^{2\mu _{j}+m+1};q ) _{\infty }}{\Gamma _{q} ( m+1 ) } \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( -m+\eta +\alpha +\frac{1}{2}k+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}

This completes the proof of the remark. □

### Theorem 3

Let $$J_{2\mu _{1}}^{ ( 2 ) } ( 2\sqrt{\delta _{1}t};q ) ,\ldots,J_{2\mu _{n}}^{ ( 2 ) } ( 2\sqrt{\delta _{n}t};q )$$ and $$f ( t ) =t^{\Delta -1}\prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 2 ) } ( 2\sqrt{\delta _{j}t};q )$$. Then, for some $$A= \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}q^{-\alpha ( \Delta -1 ) }$$, we have

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&A\prod_{j=1}^{n} \delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{-\alpha \mu _{j}}\sum _{m=0}^{ \infty }q^{m ( m+\mu _{j} ) } \frac{ ( -\delta _{j}xq^{-\alpha -k} ) ^{m} ( q^{\mu _{i}+m+1};q_{\infty } ) }{ ( q;q ) _{\infty }\Gamma _{q} ( m+k ) } \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +k+\mu _{j}+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( 1+k ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}

### Proof

Let the hypothesis of the theorem be satisfied. Then we have

$$K_{q}^{\eta ,\alpha }f ( x ) = ( 1-q ) ^{ \alpha }\sum _{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta + \alpha ) +\frac{1}{2}k ( k+1 ) } \begin{bmatrix} -\alpha \\ k\end{bmatrix} f \bigl( xq^{-\alpha -k} \bigr).$$

Therefore, in view of (18) and (3), we write

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =& ( 1-q ) ^{ \alpha }\sum _{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta + \alpha ) +\frac{1}{2}k ( k+1 ) } \frac{ ( q;q ) _{-\alpha }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}f \bigl( xq^{- \alpha -k} \bigr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \bigl( xq^{-\alpha -k} \bigr) ^{\Delta -1} \\ &{}\times \prod_{j=1}^{n}J_{2\mu _{j}}^{ ( 2 ) } \bigl( \sqrt{\delta _{j}xq^{-\alpha -k}};q \bigr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \bigl( xq^{-\alpha -k} \bigr) ^{\Delta -1} \\ &{}\times \prod_{j=1}^{n} \bigl( \delta _{j}xq^{-\alpha -k} \bigr) ^{ \mu _{j}} \sum _{m=0}^{\infty } \frac{q^{m ( m+\mu _{j} ) } ( -\delta _{j}xq^{-\alpha -k} ) ^{m}}{ ( q;q ) _{\mu _{j}+m} ( q;q ) _{m}} \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}q^{-\alpha ( \Delta -1 ) }\sum _{k=0}^{ \infty } ( -1 ) ^{q} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) -k ( \Delta -1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \\ &{}\times \prod_{j=1}^{n}\delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{- ( \alpha +k ) \mu _{j}}\sum _{m=0}^{\infty } \frac{q^{m ( m+\mu _{j} ) } ( -\delta _{j}xq^{-\alpha -k} ) ^{m}}{ ( q;q ) _{\mu _{j}+m} ( q,q ) _{m}}. \end{aligned}

Hence, it yields

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) = &\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}x^{\Delta -1}q^{-\alpha ( \Delta -1 ) } \prod_{j=1}^{n}a_{j}^{\mu _{j}}x^{\mu _{j}}q^{- ( \alpha +k ) \mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } \frac{q^{m ( m+\mu _{j} ) } ( -a_{j}xq^{-\alpha -k} ) ^{m}}{ ( q;q ) _{\mu _{j}+m} ( q;q ) _{m}} \sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) -k ( \Delta -1 ) +\mu _{j}k}}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}. \end{aligned}
(22)

By the fact $$( q;q ) _{k}=\Gamma _{q} ( 1+k )$$ and the identity

$$( \zeta ;q ) _{x}= \frac{ ( q;q ) _{\infty }}{ ( \zeta q^{x};q ) _{\infty }},$$
(23)

we write

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&A\prod_{j=1}^{n} \delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{-\alpha \mu _{j}}\sum _{m=0}^{ \infty } \frac{q^{m ( m+\mu _{j} ) } ( \delta _{j}xq^{-\alpha -k} ) ^{m}}{ ( q;q ) _{\infty } ( q;q ) _{m}} \bigl( q^{ \mu _{i}+m+1};q \bigr) _{\infty } \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) -k ( \Delta -1 ) +\mu _{j}k}}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \\ =&A\prod_{j=1}^{n}\delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{- \alpha \mu _{j}}\sum _{m=0}^{\infty }q^{m ( m+\mu _{j} ) } \frac{ ( -\delta _{j}xq^{-\alpha -k} ) ^{m} ( q^{\mu _{i}+m+1};q ) _{\infty }}{ ( q;q ) _{\infty }\Gamma _{q} ( m+k ) } \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +k+\mu _{j}+\frac{3}{2}-\Delta ) }}{\Gamma _{q} ( 1+k ) \Gamma _{q} ( 1-\alpha -k ) }. \end{aligned}

This completes the proof of the theorem. □

### Theorem 4

Let $$J_{2\mu _{1}}^{ ( 3 ) } ( 2\sqrt{q^{-1}\delta _{1}t};q ) ,\ldots,J_{2\mu _{n}}^{ ( 3 ) } ( 2\sqrt{q^{-1}\delta _{n}t};q )$$ be n q-Bessel functions and

$$f ( t ) =t^{\Delta -1}\prod_{j=1}^{n} \delta ^{ \mu _{j}}J_{2\mu _{j}}^{ ( 3 ) } \bigl( 2 \sqrt{q^{-1} \delta _{n}t};q \bigr).$$

Then we have

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&\frac{x^{\Delta -1}\Gamma _{q} ( 1-\alpha ) ( 1-q ) ^{\alpha }}{ ( q;q ) _{\infty }}\prod _{j=1}^{n}\delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{ ( -\alpha -k ) \mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } ( -1 ) ^{m} \frac{q^{m\frac{ ( m-1 ) }{2}+m ( -\alpha ) }x^{m}\delta _{j}^{m} ( q^{2\mu _{j}+m+1};q ) _{\infty }}{ ( q;q ) _{m}} \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) + ( \Delta -1 ) ( -\alpha -k ) -mk}}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( -\alpha -k ) }. \end{aligned}

### Proof

By (2) and (6), we obtain

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =& ( 1-q ) ^{ \alpha }\sum _{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta + \alpha ) +\frac{1}{2}k ( k+1 ) } \begin{bmatrix} -\alpha \\ k\end{bmatrix} f \bigl( xq^{-\alpha -k} \bigr) \\ =& ( 1-q ) ^{\alpha }\sum_{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) } \begin{bmatrix} -\alpha \\ k\end{bmatrix} \bigl( xq^{-\alpha -k} \bigr) ^{\Delta -1} \\ &{}\times \prod_{j=1}^{n}q^{\mu j}J_{2\mu _{j}}^{ ( 3 ) } \bigl( \sqrt{q^{-1}\delta _{j}xq^{-\alpha -k}};q \bigr) \\ =& ( 1-q ) ^{\alpha }\sum_{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) } \begin{bmatrix} -\alpha \\ k\end{bmatrix} \bigl( xq^{-\alpha -k} \bigr) ^{\Delta -1} \\ &{}\times \prod_{j=1}^{n}q^{\mu j} \bigl( q^{-1}\delta _{j}xq^{- \alpha -k} \bigr) ^{\mu j} \sum_{m=0}^{\infty } ( -1 ) ^{m} \frac{q^{m\frac{ ( m-1 ) }{2}} ( qq^{-1}\delta _{j}q^{-\alpha -k} ) ^{m}}{ ( q;q ) _{2\mu _{j}+m} ( q;q ) _{m}}. \end{aligned}

Equations (10), (21), and simple simplifications reveal

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x ) =&x^{\Delta -1} ( 1-q ) ^{\alpha }\sum_{k=0}^{\infty } ( -1 ) ^{k}q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) + ( \Delta -1 ) ( -\alpha -k ) } \frac{\Gamma _{q} ( 1-\alpha ) }{\Gamma _{q} ( k+1 ) \Gamma _{q} ( -\alpha -k ) } \\ &{}\times \prod_{j=1}^{n}\delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{ ( -\alpha -k ) \mu _{j}}\sum _{m=0}^{\infty } ( -1 ) ^{m} \frac{q^{m\frac{ ( m-1 ) }{2}+m ( -\alpha -k ) }x^{m}\delta _{j}^{m} ( q^{2\mu _{j}+m+1};q ) _{\infty }}{ ( q;q ) _{\infty } ( q;q ) _{m}} \\ =&\frac{x^{\Delta -1}\Gamma _{q} ( 1-\alpha ) ( 1-q ) ^{\alpha }}{ ( q;q ) _{\infty }}\prod_{j=1}^{n} \delta _{j}^{\mu _{j}}x^{\mu _{j}}q^{ ( -\alpha -k ) \mu _{j}} \\ &{}\times \sum_{m=0}^{\infty } ( -1 ) ^{m} \frac{q^{m\frac{ ( m-1 ) }{2}+m ( -\alpha ) }x^{m}\delta _{j}^{m} ( q^{2\mu _{j}+m+1};q ) _{\infty }}{ ( q;q ) _{m}} \\ &{}\times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) + ( \Delta -1 ) ( -\alpha -k ) -mk}}{\Gamma _{q} ( k+1 ) \Gamma _{q} ( -\alpha -k ) }. \end{aligned}

This completes the proof of the theorem. □

## The fractional q-integral of the power series

This section is briefly devoted to the application of the fractional q-integral to functions of a power series form. Some corollaries associated with polynomials and unit functions are also deduced.

### Theorem 5

Let $$g ( x ) =\sum_{i=0}^{\infty }r_{i}x^{i}$$ be a power series and β be a positive real number. If $$f ( x ) = ( x^{\beta -1}g ) ( x )$$, then we have

$$K_{q}^{\eta ,\alpha }f ( x ) = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k+\frac{1}{2}-i}}{\Gamma _{q} ( k ) \Gamma _{q} ( -\alpha -k ) }.$$

### Proof

Let $$g ( x ) =\sum_{i=0}^{\infty }r_{i}x^{i}$$ be a power series and β be a positive real number. From (26) it follows

\begin{aligned} K_{q}^{\eta ,\alpha }f ( x )& = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{k=0}^{\infty } ( -1 ) ^{k}\frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}f \bigl( xq^{- \alpha -k} \bigr) \\ &= \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \bigl( xq^{-\alpha -k} \bigr) ^{\beta -1} \sum_{i=0}^{\infty }r_{i} \bigl( xq^{-\alpha -k} \bigr) ^{i}. \end{aligned}
(24)

Interchanging the order of summation in (24) leads to

$$K_{q}^{\eta ,\alpha }f ( x ) = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) -k_{i}}}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}}.$$

Employing (21) indeed gives

$$K_{q}^{\eta ,\alpha }f ( x ) = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k+\frac{1}{2}-i}}{\Gamma _{q} ( k ) \Gamma _{q} ( -\alpha -k ) }.$$

Hence, the proof of the theorem is completed. □

### Corollary 6

Let $$\beta >0$$ be a real number. Then we have

$$K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1} \bigr) = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1}}{ ( 1-q ) ^{-\alpha }} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( ( \eta +\alpha ) +\frac{1}{2}k+\frac{1}{2} ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( -\alpha -k ) }.$$

This result follows from setting $$r_{0}=1$$ and $$r_{i}=0$$ for $$i=1,2,3,\ldots$$ .

### Corollary 7

We have

$$K_{q}^{\eta ,\alpha } ( 1 ) = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2} ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( -\alpha -k ) }.$$
(25)

## $$K_{q}^{\eta ,\alpha }$$ of q-generating Heines series

The basic q-generating series of the first type is defined by  as

$$_{r}\phi _{s} ( \delta _{1},\ldots,\delta _{r};b_{1},\ldots,b_{s},q, \zeta ) =\sum _{i\geq 0} \frac{ ( \delta _{1};q ) _{1},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( b_{1};q ) _{i},\ldots, ( b_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{1+s-r}\zeta ^{i},$$

where

$$\bigl( 2^{i} \bigr) =\frac{i ( i-1 ) }{2},\quad\quad r>s+1,\quad\quad q>0.$$
(26)

The basic q-generating series of the second type is given as

$$_{r}\psi _{s} ( \delta _{1},\ldots,\delta _{r};\grave{\delta }_{1},\ldots,\grave{\delta }_{s},q,\zeta ) =\sum_{i\geq 0}^{\infty } \frac{ ( \delta _{1};q ) _{1},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( \grave{\delta }_{1};q ) _{i},\ldots, ( \grave{\delta }_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{s-r}\zeta ^{i}.$$
(27)

The parameters $$b_{1},\ldots,b_{s}$$ are given so that the denominator factors in terms of the series are never zero, and the basic series terminates when one of its numerator parameters is of type $$q^{-n}$$, $$n=0,1,2,\ldots$$ .

### Theorem 8

Let β and γ be real numbers. Then, provided $$\beta >0$$, we have

\begin{aligned}& K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1} \bigr) _{r} \phi _{s} ( \delta _{1},\ldots,\delta _{r}; \grave{\delta }_{1},\ldots,\grave{\delta }_{s};q, \gamma x ) \\& \quad = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }} \\& \quad\quad {} \times \sum_{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{ \infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }. \end{aligned}

### Proof

Let β and γ be real numbers. Then, by (17), we write

\begin{aligned}& K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1} \bigr) _{r} \phi _{s} ( \delta _{1},\ldots,\delta _{r}; \grave{\delta }_{1},\ldots,\grave{\delta }_{s};q, \gamma x ) \\& \quad = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) +\frac{1}{2}k ( k+1 ) }}{ ( q;q ) _{k} ( q;q ) _{-\alpha -k}} \\& \quad\quad {}\times f \bigl( xq^{-\alpha -k} \bigr) \bigl( xq^{-\alpha -k} \bigr) _{{}}^{ \beta -1}{ }_{r}\phi _{s} \bigl( \delta _{1},\ldots,\delta _{r}; \grave{\delta }_{1},\ldots,\grave{\delta }_{s};q,\gamma xq^{-\alpha -k} \bigr) . \end{aligned}

On the other hand, we have

\begin{aligned}& _{r}\phi _{s} \bigl( \delta _{1},\ldots, \delta _{r};\grave{\delta }_{1},\ldots,\grave{ \delta }_{s};q,\gamma xq^{-\alpha -k} \bigr) \sum _{i \geq 0}^{\infty } \frac{ ( \delta _{1};q ) _{i},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( \grave{\delta }_{1};q ) _{i},\ldots, ( \grave{\delta }_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{s-r} \\& \quad {} \times \bigl( \gamma xq^{-\alpha -k} \bigr) ^{i}=\sum _{i \geq 0}^{\infty }r_{i}x^{i}, \end{aligned}

where

$$r_{i}= \frac{ ( \delta _{1};q ) _{i},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( \grave{\delta }_{1};q ) _{i},\ldots, ( \grave{\delta }_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{s-r}\gamma ^{i}q^{ ( -\alpha -k ) i}.$$
(28)

Therefore, by Theorem 5 we get

\begin{aligned} K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1} \bigr) _{r} \phi _{s} ( \delta _{1},\ldots,\delta _{r}; \grave{\delta }_{1},\ldots,\grave{\delta }_{s};q, \gamma x ) =& \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{ \infty }r_{i}q^{-\alpha i}x^{i} \\ &{} \times \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }. \end{aligned}
(29)

This completes the proof of the theorem. □

### Theorem 9

Let $$\beta >0$$ and r be real numbers. Then we have

\begin{aligned} K_{q}^{\eta ,\alpha } \bigl( x_{r}^{\beta -1}\psi _{s} ( \delta _{1},\ldots, \delta _{r}; \grave{\delta }_{1},\ldots,\grave{\delta }_{s};q,\gamma x ) \bigr) =&\frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \\ &{} \times \sum_{k=0}^{\infty } ( -1 ) ^{k}\frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }. \end{aligned}

### Proof

By taking into account (20), we write

\begin{aligned} K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1}{ }_{r}\psi _{s} \bigr) =& \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum _{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha ) + \frac{1}{2}k+\frac{1}{2}}}{ ( q;q ) _{k} ( q;q ) _{-\alpha ,k}}f \bigl( xq^{-\alpha -k} \bigr) _{{}}^{\beta -1} \\ &{} \times f \bigl( xq^{-\alpha -k} \bigr) _{{}}^{\beta -1}{ } _{r} \psi _{s} \bigl( \delta _{1},\ldots, \delta _{r};\grave{\delta }_{1},\ldots,\grave{ \delta }_{s};q^{-\alpha -k} \bigr) . \end{aligned}
(30)

However,

\begin{aligned} _{r}\psi _{s} \bigl( \delta _{1},\ldots, \delta _{r};\grave{\delta }_{1},\ldots,\grave{ \delta }_{s};q^{-\alpha -k} \bigr) =&\sum _{i\geq 0}^{ \infty }\frac{ ( \delta _{1};q ) _{i},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( \grave{\delta }_{1};q ) _{i},\ldots, ( \grave{\delta }_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{s-r} \\ & {} \times \bigl( \gamma xq^{-\alpha -k} \bigr) ^{i} \\ =&\sum_{i\geq 0}^{\infty }r_{i}x^{i}, \end{aligned}

where

$$r_{i}= \frac{ ( \delta _{1};q ) _{i},\ldots, ( \delta _{r};q ) _{i}}{ ( q;q ) _{i}, ( \grave{\delta }_{1};q ) _{i},\ldots, ( \grave{\delta }_{s};q ) _{i}} \bigl( ( -1 ) ^{i}q^{ ( 2^{i} ) } \bigr) ^{s-r}\gamma ^{i}q^{ ( -\alpha -k ) i}.$$
(31)

Hence, by Theorem 5 it follows

$$K_{q}^{\eta ,\alpha } \bigl( x^{\beta -1}{ }_{r}\psi _{s} \bigr) = \frac{q^{-\alpha \beta +\alpha }x^{\beta -1} ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }.$$

This completes the proof of the theorem. □

### Corollary 10

Let γ be a real number. Then we have

$$K_{q}^{\eta ,\alpha } \bigl( E_{q} ( \gamma x ) \bigr) = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{ \infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }.$$

### Proof

By setting $$\beta =0$$, $$r=0$$, and $$s=0$$, the result easily follows from Theorem 8. The proof is completed. □

### Corollary 11

Let γ be a real number. Then we have

$$\bigl( K_{q}^{\eta ,\alpha }e_{q} ( \gamma x ) \bigr) = \frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{\infty }r_{i}q^{-\alpha i}x^{i} \sum_{k=0}^{ \infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k+\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }.$$

### Proof

By setting $$\beta =1$$, $$r=0$$, and $$s=0$$, Theorem 8 completes the proof of the corollary. □

The proof of the following corollary is straightforward. Details are therefore deleted.

### Corollary 12

Let γ be a real number. Then we have

\begin{aligned} ( i ) K_{q}^{\eta ,\alpha } \bigl( \sinh _{q} ( \gamma x ) \bigr) =&K_{q}^{\eta ,\alpha } \biggl( \frac{E_{q} ( \gamma x ) -E_{q} ( -\gamma x ) }{2} \biggr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{\infty }r_{i}q^{-\alpha i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}k-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }x^{i} \frac{ ( 1+ ( -1 ) ^{i+1} ) }{2}. \end{aligned}
\begin{aligned} ( \mathit{ii} ) K_{q}^{\eta ,\alpha } \bigl( \cosh _{q} ( \gamma x ) \bigr) =&K_{q}^{\eta ,\alpha } \biggl( \frac{E_{q} ( \gamma x ) -E_{q} ( -\gamma x ) }{2} \biggr) \\ =&\frac{ ( q;q ) _{-\alpha }}{ ( 1-q ) ^{-\alpha }}\sum_{i=0}^{ \infty }r_{i}q^{-\alpha i} \sum_{k=0}^{\infty } ( -1 ) ^{k} \frac{q^{k ( \eta +\alpha +\frac{1}{2}-i ) }}{\Gamma _{q} ( k ) \Gamma _{q} ( _{-\alpha -k} ) }x^{i} \frac{ ( 1+ ( -1 ) ^{i} ) }{2}. \end{aligned}

## References

1. Jackson, F.H.: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)

2. Strominger, A.: Information in black hole radiation. Phys. Rev. Lett. 71, 3743–3746 (1993)

3. Youm, D.: q-Deformed conformal quantum mechanics. Phys. Rev. D 62, 095009 (2000)

4. Lavagno, A., Swamy, P.N.: q-Deformed structures and nonextensive statistics: a comparative study. Physica A 305(1–2), 310–315 (2002)

5. Hahn, W.: Beitrage Zur Theorie Der Heineschen Reihen, die 24 Integrale der hypergeometrischen q-diferenzengleichung, das q-Analog on der Laplace transformation. Math. Nachr. 2, 340–379 (1949)

6. IsmailM, E.H.: The zeros of basic Bessel functions, the functions $$J_{v+ax}(x)$$, and associated orthogonal polynomials. J. Math. Anal. Appl. 86(1), 1–19 (1982)

7. Jackson, F.H.: The application of basic numbers to Bessel’s and Legendre’s functions. Proc. Lond. Math. Soc. 2(1), 192–220 (1905)

8. Hahn, W.: Die mechanische deutung einer geometrischen differenzengleichung. Z. Angew. Math. Mech. 33, 270–272 (1953)

9. Exton, H.: A basic analogue of the Bessel-Clifford equation. Jñānābha 8, 49–56 (1978)

10. Salem, A.: Some applications of fractional q-calculus. J. Fract. Calc. Appl. 2(4), 1–11 (2012)

11. Ucar, F.: q-Sumudu transforms of q-analogues of Bessel functions. Sci. World J. 2014, 327019 (2014)

12. Majeed, A., Kamran, M., Abbas, M., Singh, J.: An efficient numerical technique for solving time-fractional generalized Fisher’s equation. Front. Phys. 8, 293 (2020)

13. Akram, T., Abbas, M., Ali, A., Iqbal, A., Baleanu, D.: A numerical approach of a time fractional reaction–diffusion model with a non-singular kernel. Symmetry 12, 1653 (2020)

14. Akram, T., Muhammad, A., Azhar, I., Dumitru, B., Jihad, H.: Novel numerical approach based on modified extended cubic B-spline functions for solving non-linear time-fractional telegraph equation. Symmetry 12, 1154 (2020)

15. Akram, T., Abbas, M., Iqbal, A., Baleanu, D., Asad, J.: Numerical treatment of time-fractional Klein–Gordon equation using redefined extended cubic B-spline functions. Front. Phys. 8, 288 (2020)

16. Majeed, A., Kamran, M., Abbas, M., Md, M.: An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin–Bona–Mahony–Burger model. Phys. Scr. 96(8), 084002 (2021)

17. Al-Omari, S.: Certain results related to the N-transform of certain class of functions and differential operators. Adv. Differ. Equ. 2018, 7 (2018)

18. Al-Omari, S.K.Q.: On the application of natural transforms. Int. J. Pure Appl. Math. 84, 729–744 (2013)

19. Al-Omari, S., Baleanu, D.: Quaternion Fourier integral operators for spaces of generalized quaternions. Math. Methods Appl. Sci. 41, 9477–9484 (2018)

20. Al-Omari, S., Kilicman, A.: An estimate of Sumudu transform for Boehmians. Adv. Differ. Equ. 2013, 77 (2013)

21. Al-Omari, S., Agarwal, P., Choi, J.: Real covering of the generalized Hankel-Clifford transform of fox kernel type of a class of Boehmians. Bull. Korean Math. Soc. 52(5), 1607–1619 (2015)

22. Al-Omari, S., Agarwal, P.: Some general properties of a fractional Sumudu transform in the class of Boehmians. Kuwait J. Sci. 43(2), 206–220 (2016)

23. Kac, V., Cheung, P.: Quantum Calculus. Springer, Berlin (2001)

24. Al-Omari, S.: The q-Sumudu transform and its certain properties in a generalized q-calculus theory. Adv. Differ. Equ. 2021, 10 (2021)

25. Salem, A., Ucar, F.: The q-analogue of the $$E_{(2;1)}$$ g-transform and its applications. Turk. J. Math. 40(1), 98–107 (2016)

26. Serkan, A., Ugur, D., Mehmet, A.: On weighted q-Daehee polynomials with their applications. Indag. Math. 30(2), 365–374 (2019)

27. Vyas, V., Al-Jarrah, A., Purohit, S., Araci, S., Nisar, K.: q-Laplace transform for product of general class of q-polynomials and q-analogue of L-function. J. Inequal. Spec. Funct. 11(3), 21–28 (2020)

28. Al-Omari, S.: q-Analogues and properties of the Laplace-type integral operator in the quantum calculus theory. J. Inequal. Appl. 2020, 203 (2020)

29. Al-Omari, S.K.: 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)

30. Al-Omari, S.K.Q., Baleanu, D., Purohit, S.D.: Some results for Laplace-type integral operator in quantum calculus. Adv. Differ. Equ. 2018, 124 (2018)

31. Chandak, S., Suthar, D.L., Al-Omari, S., Gulyaz-Ozyurt, S.: Estimates of classes of generalized special functions and their application in the fractional $$(k,s)$$-calculus theory. J. Funct. Spaces 2021 (2021, to appear)

32. Al-Omari, S.: Estimates and properties of certain q-Mellin transform on generalized q-calculus theory. Adv. Differ. Equ. 2021, 233 (2021)

33. Al-Omari, S.: On q-analogues of Mangontarum transform of some polynomials and certain class of H-functions. Nonlinear Stud. 23(1), 51–61 (2016)

34. Albayrak, D., Purohit, S.D., Ucar, F.: On q-Sumudu transforms of certain q-polynomials. Filomat 27(2), 413–429 (2013)

35. Al-Salam, W.A., Verma, A.: A fractional Leibniz q-formula. Pac. J. Math. 60, 1–9 (1975)

36. Al-Salam, W.A.: Some fractional q-integrals and q-derivatives, Proc. Edinb. Math. Soc. 15, 135–140 (1966)

37. Agrawal, R.P.: Certain fractional q-integrals and q-derivatives. Math. Proc. Camb. Philos. Soc. 66, 365–370 (1969)

38. Atici, F.M., Eloe, P.W.: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14(3), 333–344 (2007)

39. Rajkovic, P.M., Marinkovi, S.D., Stankovi, M.S.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)

40. 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)

41. Al-Omari, S.: On a q-Laplace–type integral operator and certain class of series expansion. Math. Methods Appl. Sci. 44(10), 8322–8332 (2021)

## Acknowledgements

Authors would like to thank Springer Nature for their support.

## Funding

No funding sources to be declared.

## Author information

Authors

### Contributions

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

### Corresponding author

Correspondence to Shrideh Al-Omari.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions 