| Summary: | 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.
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