Generating Functions for Spherical Harmonics and Spherical Monogenics

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in \({\mathbb{R}^{m}}\) . Here spherical monogenics are polynomial solutions of the Dirac equation in \({\mathbb{R}^{m}}\) . In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m–1. Hence we can find closed formulæ of generating functions in \({\mathbb{R}^{m}}\) by induction on the dimension m.