Weighted Sylvester sums on the Frobenius set

Abstract

Let $a$ and $b$ be relatively prime positive integers. In this paper the weighted sum ${\sum }_{n\in \mathrm{NR}(a,b)}{\lambda }^{n-1}{n}^m$ is given explicitly or in terms of the Apostol-Bernoulli numbers, where $m$ is a nonnegative integer, and $\mathrm{NR}(a,b)$ denotes the set of positive integers nonrepresentable in terms of $a$ and $b$.

Irish Math. Soc. Bull. (to appear)