Extension of the Bernoulli and Eulerian Polynomials of Higher Order and Vector Partition Function

Abstract

Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W(\mathbf{s},\mathbf{D})$, i.e., a number of integer solutions to a linear system $\mathbf{x}\ge 0,\mathbf{Dx}=\mathbf{s}$. It is shown that $W(\mathbf{s},\mathbf{D})$ can be expressed through the vector Bernoulli polynomials of higher order.

18 pages