Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

Explicit expressions for restricted partition function $W(s,{\mathbf{d}}^m)$ and its quasiperiodic components $W_j(s,{\mathbf{d}}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\mathbf{d}}^m=\{d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_m\}$ 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.