Generalized Bessel functions for p-radial functions

Suppose that d N and p > 0. In this paper, we study the generalized Bessel functions for the surface 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 W d , which is the symmetry group of the # p unit sphere. By means of this symmetry under W d , 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 . 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. 2000 AMS subj. class.: 33C10, 42B10, 44A15, 62H10 Keywords: radial functions, complete symmetric functions, monomial symmetric functions, Fourier-Bessel transform, divided di#erences, characteristic functions, multivariate symmetric distributions, reflection symmetry, finite reflection groups, Gelfand pairs