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 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.

Automated summary

Using Mellin-Barnes contour integrals, we propose a q-extension of the several variable Aleph-function. We then present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. By using the q-analogue of the Leibniz rule, we provide a formula expressed in terms of an infinite series for the q-extended several variable Aleph-function. The general nature of the formulas presented in this article allows for the reduction to yield identities involving q-extended simpler special functions, such as the q-extended I-function, the q-extended H-function, and the q-extended Meijer's G-function. These results are believed to have applications in quantum mechanics.