Cubic Harmonics and Bernoulli Numbers

Details Cubic Harmonics and Bernoulli Numbers Cubic Harmonics and Bernoulli Numbers

2011-10-25

arxiv.org/abs/1110.5540v1

Mathematics

Combinatorics

18 pages, 3 figures

Scientific paper

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.

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