$(p,q)$-Whittaker function and associated properties and formulas

TL;DR

This paper introduces an extended $(p,q)$-Whittaker function based on the $(p,q)$-confluent hypergeometric function, exploring its properties, formulas, and connections to classical Whittaker functions.

Contribution

It presents a novel extension of the $(p,q)$-Whittaker function and derives its integral, transformation, and differential formulas, linking it to existing Whittaker functions.

Findings

Derived integral representations of the extended $(p,q)$-Whittaker function.

Established transformation and differential formulas for the new function.

Connected the extended function to classical Whittaker functions.

Abstract

Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $\Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an extended $(p,q)$-Whittaker function by using the function $\Phi_{p,q}$ and establish its various properties and associated formulas such as integral representations, some transformation formulas and differential formulas. Relevant connections of the results presented here With those involving relatively simple Whittaker functions are also pointed out.