Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order
Boris Y. Rubinstein, Leonid G. Fel
Abstract
Explicit expressions for restricted partition function W(s, dm) and its quasiperiodic components Wj(s, dm) (called Sylvester waves) for a set of positive integers dm = \d1, d2, ..., dm\ 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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