An Explicit Formula for Restricted Partition Function through Bernoulli Polynomials

Explicit expressions for restricted partition function $W(s,{\mathbf{d}}^m)$ and its quasiperiodic components $W_j(s,{\mathbf{d}}^m)$ (called 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 polynomials of higher order with periodic coefficients.