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

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.