  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.
