On extension of the $_{r}R_{s,q}(\alpha,\beta,z)$ function and their $q$-calculus

TL;DR

This paper explores new properties, relations, and integral representations of the $q$-analogue of the $_{r}R_{s,q}$ function, including fractional derivatives and integrals within $q$-calculus.

Contribution

It introduces novel properties and representations of the $q$-analogue $_{r}R_{s,q}$ function, expanding the theoretical framework of $q$-calculus.

Findings

Derived new integral representations of $_{r}R_{s,q}$ functions

Established fractional $q$-derivative and $q$-integral operators

Presented examples illustrating theoretical results

Abstract

In this article, we investigate and establish some properties including analytic properties, contiguous relations, differential properties, differential operators, an expansion formula, and simple integrals, integral operators, some fractional integral properties, some new integral representations, the Riemann Liouville fractional $q$-derivative and $q$-integral operators of $q$-analogue of various basic $_{r}R_{s,q}$ function by using technique of $q$-calculus. Certain interesting consequences of the theorem are also discussed by considering some examples.