Cubic harmonics and Bernoulli numbers
Copyright policy )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.
18 pages, 3 figures
- Hokkeido University Japan
- Hokkaido Bunkyo University Japan
- Hokkaido University Japan
Partitions, Invariant theory, 413, 52B15, 20F55, 11B68, polyhedral harmonics, Harmonic, subharmonic, superharmonic functions in higher dimensions, Invariant differential equations, \(n\)-dimensional polytopes, Theoretical Computer Science, Polyhedral harmonics, generating functions, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Reflection groups, Young diagrams, invariant differential equations, Cube, Other designs, configurations, invariant theory, Generating functions, Computational Theory and Mathematics, reflection groups, partitions, Association schemes, strongly regular graphs, Combinatorics (math.CO), Bernoulli and Euler numbers and polynomials, cube, Bernoulli numbers
Partitions, Invariant theory, 413, 52B15, 20F55, 11B68, polyhedral harmonics, Harmonic, subharmonic, superharmonic functions in higher dimensions, Invariant differential equations, \(n\)-dimensional polytopes, Theoretical Computer Science, Polyhedral harmonics, generating functions, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Reflection groups, Young diagrams, invariant differential equations, Cube, Other designs, configurations, invariant theory, Generating functions, Computational Theory and Mathematics, reflection groups, partitions, Association schemes, strongly regular graphs, Combinatorics (math.CO), Bernoulli and Euler numbers and polynomials, cube, Bernoulli numbers
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