# Relations among spheroidal and spherical harmonics

## TL;DR

This paper explores the relationship between spheroidal and spherical harmonics, introducing contragenic functions in spheroidal domains and establishing their properties and interrelations across different eccentricities.

## Contribution

It defines contragenic functions in spheroidal domains and provides computational formulas linking harmonic bases across domains with varying eccentricities.

## Key findings

- Existence of nontrivial contragenic functions common to all spheroidal eccentricities
- Established computational formulas relating harmonic bases between spheroids and spheres
- Contragenic functions depend on the domain and are not local properties

## 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.