Generalized holomorphic Szegö 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 R4R4), and are generally assumed to be nullsolutions of the well known Moisil–Théodoresco system. In this paper, we show that these polynomial functions play an important role in defining the Szegö kernel function over the surface of 3D (prolate) spheroids. As a concrete application, we prove an explicit expression of the monogenic Szegö kernel function over 3D (prolate) spheroids and present two numerical examples.