A generalization of certain associated Bessel functions in connection with a group of shifts

TL;DR

This paper introduces a new class of special functions related to Bessel functions, derived from integral kernels associated with group representations, and explores their properties and connections to classical functions.

Contribution

It generalizes certain associated Bessel functions through integral kernels linked to group actions, establishing new formulas and orthogonality relations.

Findings

Derived formulas involving Whittaker and Weber functions

Identified parameter values matching Bessel function variants

Established orthogonality relations for related special functions

Abstract

Considering the kernel of an integral operator intertwining two realizations of the group of motions of the pseudo-Euclidian space, we derive two formulas for series containing Whittaker's functions or Weber's parabolic cylinder functions. We can consider this kernel as a special function. Some particular values of parameters involved in this special function are found to coincide with certain variants of Bessel functions. Using these connections, we also establish some analogues of orthogonality relations for Macdonald and Hankel functions.