On some identities for confluent hypergeometric functions and Bessel functions

TL;DR

This paper introduces new integral representations and summation formulas for confluent hypergeometric and Bessel functions, enhancing the understanding of their properties and interrelations in mathematical analysis.

Contribution

It presents novel integral representations of the Whittaker function and summation formulas for Kummer's confluent hypergeometric functions, linking known and new results.

Findings

New integral representation of the Whittaker function

Relevant summation formula for Kummer's confluent hypergeometric functions

Connections to known and new identities in special functions

Abstract

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results.