Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Abstract

Explicit expressions for restricted partition function W(s,d m ) and its quasiperiodic components W j (s,d m ) (called Sylvester waves) for a set of positive integers d m = {d 1, d 2, ..., d m } are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian 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 finite groups is discussed.