Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

2003/04/23 by Boris Y. Rubinstein, Rubinstein, Boris Y., Leonid G. Fel +1 · 1 citation
Mathematics · #05A17 (Primary) #11P81 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:05A17 #msc:11P81

paper · pdf · doi:10.48550/arxiv.math/0304356

15 pages, 2 figures, references added, submitted to The Ramanujan Journal

arxiv created 2003/05/16 · arxiv updated 2009/11/30

Abstract

Explicit expressions for restricted partition function W(s,\bf d^m) and its quasiperiodic components W_j(s,\bf d^m) (called \em Sylvester waves) for a set of positive integers \bf d^m = \d₁, d₂, ..., d_m\ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.

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