Product Formulas and Convolutions for Angular and Radial Spheroidal Wave Functions

Abstract

Product formulas for angular spheroidal wave functions on $[0,\pi ]$ and for radial spheroidal wave functions on $[0,\infty )$ are presented, which generalize results for the ultraspherical polynomials and functions as well as for the Mathieu functions. Although these functions cannot be given in closed form, the kernels of the product formulas are represented in an explicit, and surprisingly simple way in terms of Bessel functions so that the exact range of positivity can easily be read off. The formulas are used to introduce two families of convolution structures on $[0,\pi ]$ and $[0,\infty )$, many of which provide new hypergroups. We proceed from the fact that the spheroidal wave functions are eigenfunctions of Sturm-Liouville equations of confluent Heun type and employ a partial differential equation technique based on Riemann’s integration method.