Generalized holomorphic Szegoe kernel in 3D spheroids

Monogenic orthogonal polynomials over 3D prolate spheroids were previously introduced and shown to have some remarkable properties. In particular, the underlying functions take values in the quaternions (identified with R super(4)), and are generally assumed to be nullsolutions of the well known Moisil-Theodoresco system. In this paper, we show that these polynomial functions play an important role in defining the Szegoe kernel function over the surface of 3D (prolate) spheroids. As a concrete application, we prove an explicit expression of the monogenic Szego kernel function over 3D (prolate) spheroids and present two numerical examples.