Certain <i>q</i>-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 <i>q</i>-analogue (<i>q</i>-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional <i>q</i>-integral and <i>q</i>-differential formulae for the <i...
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MDPI AG
2023-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/1/51 |
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| author | Dinesh Kumar Frédéric Ayant Norbert Südland Junesang Choi |
| author_facet | Dinesh Kumar Frédéric Ayant Norbert Südland Junesang Choi |
| author_sort | Dinesh Kumar |
| collection | DOAJ |
| description | Using Mellin-Barnes contour integrals, we aim at suggesting a <i>q</i>-analogue (<i>q</i>-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional <i>q</i>-integral and <i>q</i>-differential formulae for the <i>q</i>-extended several variable Aleph-function. Using the <i>q</i>-analogue of the Leibniz rule for the fractional <i>q</i>-derivative of a product of two basic functions, we also provide a formula for the <i>q</i>-extended several variable Aleph-function, which is expressed in terms of an infinite series of the <i>q</i>-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 <i>q</i>-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse <i>q</i>-extended special functions, for example, the <i>q</i>-extended <i>I</i>-function, the <i>q</i>-extended <i>H</i>-function, and the <i>q</i>-extended Meijer’s <i>G</i>-function. The results presented here are hoped and believed to find some applications, in particular, in quantum mechanics. |
| first_indexed | 2024-03-09T13:36:26Z |
| format | Article |
| id | doaj.art-cc674bb2757046c4bc4ea0732730ab02 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T13:36:26Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-cc674bb2757046c4bc4ea0732730ab022023-11-30T21:11:38ZengMDPI AGAxioms2075-16802023-01-011215110.3390/axioms12010051Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-FunctionDinesh Kumar0Frédéric Ayant1Norbert Südland2Junesang Choi3Department of Applied Sciences, College of Agriculture-Jodhpur, Agriculture University Jodhpur, Jodhpur 342304, IndiaCollége Jean L’herminier, Allée des Nymphéas, 83500 La Seyne-sur-Mer, FranceAage GmbH, Röntgenstraße 24, 73431 Aalen, Baden-Württemberg, GermanyDepartment of Mathematics, Dongguk University, Gyeongju 38066, Republic of KoreaUsing Mellin-Barnes contour integrals, we aim at suggesting a <i>q</i>-analogue (<i>q</i>-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional <i>q</i>-integral and <i>q</i>-differential formulae for the <i>q</i>-extended several variable Aleph-function. Using the <i>q</i>-analogue of the Leibniz rule for the fractional <i>q</i>-derivative of a product of two basic functions, we also provide a formula for the <i>q</i>-extended several variable Aleph-function, which is expressed in terms of an infinite series of the <i>q</i>-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 <i>q</i>-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse <i>q</i>-extended special functions, for example, the <i>q</i>-extended <i>I</i>-function, the <i>q</i>-extended <i>H</i>-function, and the <i>q</i>-extended Meijer’s <i>G</i>-function. The results presented here are hoped and believed to find some applications, in particular, in quantum mechanics.https://www.mdpi.com/2075-1680/12/1/51Mellin-Barnes contour integralsfractional calculusfractional <i>q</i>-calculus<i>q</i>-several variable Aleph-function<i>q</i>-several variable <i>I</i>-function<i>q</i>-Leibniz rule |
| spellingShingle | Dinesh Kumar Frédéric Ayant Norbert Südland Junesang Choi Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function Axioms Mellin-Barnes contour integrals fractional calculus fractional <i>q</i>-calculus <i>q</i>-several variable Aleph-function <i>q</i>-several variable <i>I</i>-function <i>q</i>-Leibniz rule |
| title | Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function |
| title_full | Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function |
| title_fullStr | Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function |
| title_full_unstemmed | Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function |
| title_short | Certain <i>q</i>-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function |
| title_sort | certain i q i analogue of fractional integrals and derivatives involving basic analogue of the several variable aleph function |
| topic | Mellin-Barnes contour integrals fractional calculus fractional <i>q</i>-calculus <i>q</i>-several variable Aleph-function <i>q</i>-several variable <i>I</i>-function <i>q</i>-Leibniz rule |
| url | https://www.mdpi.com/2075-1680/12/1/51 |
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