# Spherical polyharmonics and Poisson kernels for polyharmonic functions

## TL;DR

This paper introduces spherical polyharmonics, extending spherical harmonics, and constructs Poisson kernels for polyharmonic functions on rotated balls, linking classical and holomorphic function kernels.

## Contribution

It develops the theory of spherical polyharmonics and zonal polyharmonics, and derives Poisson kernels for polyharmonic functions using Gegenbauer polynomials.

## Key findings

- Representation of Poisson kernels in terms of Gegenbauer polynomials
- Connection between classical harmonic Poisson kernels and polyharmonic kernels
- Relation of polyharmonic Poisson kernels to the Cauchy-Hua kernel

## Abstract

We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.