An orthogonal system of monogenic polynomials over prolate spheroids in R3

The object of this paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Moisil–Théodoresco system over prolate spheroids in ${‐{R}}^3$ . This will be done in the spaces of square integrable functions over $‐{H}$ . A big breakthrough is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. The representations of these polynomials are given explicitly, ready to be implemented on a computer. In addition, we show a corresponding orthogonality of the same polynomials over the surface of the spheroids with respect to a suitable weight function.