Integral identities derived from the complex Funk-Hecke formula

Abstract

In this paper we derive integral identities involving both the unit sphere and the unit disk or subsets thereof. In addition these identities lead to a prototype of the Funk-Hecke formula for subspheres embedded in ${\Omega }_{2q}$. The technique requires the use of the complex Funk-Hecke formula, where eigenvalues of the integral operator generated by a bizonal kernel on the unit sphere ${\Omega }_{2q}$ of ${\mathbb{C}}^q$ are given by an integral on the closed unit disk $B_q$ of ${\mathbb{C}}^{q-1}$.