An Explicit Formula for the Monogenic Szego Kernel Function on 3D Spheroids

Complete orthogonal systems of monogenic polynomials over 3D prolate spheroids have been previously introduced and shown to have some important properties. In particular, the underlying functions take on values in the quaternion algebra (identified with R-4), and are nullsolutions of the well known Moisil-Theodoresco system. In this paper we introduce a new complete orthogonal system of monogenic polynomials as solutions of this system for the space exterior to 3D prolate spheroids. Additionally, we show how monogenic polynomials for the interior and exterior of a spheroid look like once their values on the surface are prescribed. With the help of these polynomials an explicit expression of the monogenic Szego kernel function over the surface of 3D spheroids is given.