Relations among spheroidal and spherical harmonics

Abstract

A contragenic function in a domain $\Omega \subseteq {\mathbf{R}}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic function, which is orthogonal in $L^2(\Omega )$ to all monogenic functions and their conjugates. The notion of contragenicity depends on the domain and thus is not a local property, in contrast to harmonicity and monogenicity. For spheroidal domains of arbitrary eccentricity, we relate standard orthogonal bases of harmonic and contragenic functions for one domain to another via computational formulas. This permits us to show that there exist nontrivial contragenic functions common to the spheroids of all eccentricities.