Certain q-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function

Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of the Leibniz r...

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Bibliographic Details
Published in:Axioms Vol. 12; no. 1; p. 51
Main Authors: Kumar, Dinesh, Ayant, Frédéric, Südland, Norbert, Choi, Junesang
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.01.2023
Subjects:
Calculus
Fractional calculus
fractional q-calculus
Identities
Infinite series
Integrals
Mellin-Barnes contour integrals
q-Leibniz rule
q-several variable Aleph-function
q-several variable I-function
Quantum mechanics
ISSN:2075-1680, 2075-1680
Online Access:Get full text
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Summary:Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of the Leibniz rule for the fractional q-derivative of a product of two basic functions, we also provide a formula for the q-extended several variable Aleph-function, which is expressed in terms of an infinite series of the q-extended several variable Aleph-function. Since the three main formulas presented in this article are so general, they can be reduced to yield a number of identities involving q-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse q-extended special functions, for example, the q-extended I-function, the q-extended H-function, and the q-extended Meijer’s G-function. The results presented here are hoped and believed to find some applications, in particular, in quantum mechanics.
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12010051