Cubic Harmonics and Bernoulli Numbers

TL;DR

This paper explicitly determines functions with the mean value property on an n-dimensional cube, linking invariant theory, differential equations, and Bernoulli numbers to solve a classical problem in polyhedral harmonics.

Contribution

It provides an explicit solution to the mean value property problem for cubes using invariant polynomials, Young diagrams, and Bernoulli numbers, connecting several mathematical theories.

Findings

Explicit formulas for cube harmonic functions

Reduction of the problem to invariant polynomial basis

Use of Bernoulli numbers in recursive solution

Abstract

The functions satisfying the mean value property for an n-dimensional cube are determined explicitly. This problem is related to invariant theory for a finite reflection group, especially to a system of invariant differential equations. Solving this problem is reduced to showing that a certain set of invariant polynomials forms an invariant basis. After establishing a certain summation formula over Young diagrams, the latter problem is settled by considering a recursion formula involving Bernoulli numbers. Keywords: polyhedral harmonics; cube; reflection groups; invariant theory; invariant differential equations; generating functions; partitions; Young diagrams; Bernoulli numbers.