On the properties of the $(p,\nu)$-extension of the Whittaker function $M_{\kappa,\mu}(z)$

TL;DR

This paper introduces a new $(p, u)$-extension of the Whittaker function using an extended hypergeometric function, and explores its properties including integral representations, transformations, asymptotics, and inequalities.

Contribution

It presents the first $(p, u)$-extension of the Whittaker function and derives its fundamental properties, expanding the function's theoretical framework.

Findings

Derived integral representations of the extended Whittaker function.

Established a $(p, u)$-analogue of Kummer's transformation.

Provided asymptotic and inequality results for the extended function.

Abstract

In this paper, we obtain a $(p,\nu)$-extension of the Whittaker function $M_{\kappa,\mu}(z)$ by using the extended confluent hypergeometric function of the first kind $\Phi_{p,\nu}(b;c;z)$ introduced in Parmar et al. [J. Classical Anal. 11 (2017) 81--106]. Also, we derive some of the main properties of this function, namely several integral representations, a summation formula, the analogue of Kummer's transformation formula, an asymptotic representation, the Mellin transform, a differential formula and some inequalities.