Generalized Bessel Functions for p-Radial Functions

Abstract

Suppose that \(d\in{\Bbb N}\) and p > 0. In this paper we study the generalized Bessel functions for the surface \(\{{\bf v}\in{\Bbb R}^d:|{\bf v}|_p=1\}\), introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the action of the hyperoctahedral group Wd, which is the symmetry group of the \(\ell_p\) unit sphere. By means of this symmetry under Wd, we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups. For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order (d - 2)/2. For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of summability problems for 1-radial functions, and we show how their results may be framed within our more general context.