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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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
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Abstract	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.
AbstractList	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.
Author	Kumar, Dinesh Ayant, Frédéric Choi, Junesang Südland, Norbert
Author_xml	– sequence: 1 givenname: Dinesh orcidid: 0000-0001-5415-1777 surname: Kumar fullname: Kumar, Dinesh – sequence: 2 givenname: Frédéric surname: Ayant fullname: Ayant, Frédéric – sequence: 3 givenname: Norbert surname: Südland fullname: Südland, Norbert – sequence: 4 givenname: Junesang orcidid: 0000-0002-7240-7737 surname: Choi fullname: Choi, Junesang
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Snippet	Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann...
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SubjectTerms	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
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Title	Certain q-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function
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