A study on unification of generalized hypergeometric function and Mittag-Leffler function with certain integral transforms of generalized basic hypergeometric function

Abstract

This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between ${}_{2}{{R}_{1}}^{\upsilon }(\mathfrak{z})$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by ${}_{2}{{R}_{1}}^{\upsilon ,q}(\mathfrak{z})$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.

Keywords

• basic hypergeometric functions in one variable ${}_r \phi _s$
• q-calculus and related topics
• mittag-leffler functions and generalizations
• integral transforms of special functions