Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

Details Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

2003-04-23

arxiv.org/abs/math/0304356v2

Mathematics

Number Theory

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

Scientific paper

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 ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ 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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