Cubic Harmonics and Bernoulli Numbers

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.