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 [1]. 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 [2], conformal quantum mechanics [3], and nuclear physics of high energy [4]. 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 [1]
$$ \int _{0}^{\xi }f ( t ) \,d_{q}t=\xi ( 1-q ) \sum_{j=0}^{\infty }q^{j}f \bigl( \xi q^{j} \bigr) $$
(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 [5] and Ismail [6], is defined by [7] 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 [7]
$$ J_{\mu }^{ ( 2 ) } ( \xi ;q ) = \biggl( \frac{\xi }{2} \biggr) ^{\mu }\sum_{j=0}^{\infty } \frac{q^{j ( j+\mu ) } ( \frac{-\xi }{4}^{2} ) ^{j}}{ ( q;q ) _{\mu +j} ( q;q ) _{j}}, \quad \xi \in \mathbb{C} . $$
(5)
Hahn [8] and Exton [9] 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 [11]
$$ \Gamma _{q} ( t ) = \int _{0}^{\frac{1}{1-q}}x^{t-1}E_{q} ( -qx ) \,d_{q}x\quad \text{and}\quad \hat{\Gamma }_{q} ( t ) = \int _{0}^{\infty }x^{t-1}e_{q} ( -x ) \,d_{q}x. $$
(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 [12–16]. 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 [17–22] and [23–34]. 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., [37–39]. 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.