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.
Highlights
The theory of q-calculus is an old subject centered on the idea of deriving q-analogous results without using limits
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
Jackson defines the q-analogue of the Bessel function of the second type as [7]
Summary
The theory of q-calculus is an old subject centered on the idea of deriving q-analogous results without using limits. Jackson defines the q-analogue of the Bessel function of the second type as [7] The q-analogue of the exponential function of the second type is given by Whereas the q-analogue of the exponential function of the first type is given by Perhaps due to explosion in research within the fractional calculus setting, new developments in the theory of fractional q-difference calculus, the q-analogues of the integral and the differential fractional operator properties were made, see, e.g., [37,38,39].
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