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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Vydáno v:Axioms Ročník 12; číslo 1; s. 51
Hlavní autoři: Kumar, Dinesh, Ayant, Frédéric, Südland, Norbert, Choi, Junesang
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.01.2023
Témata:
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
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12010051