Weighted Sylvester sums on the Frobenius set in more variables

Abstract

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Let ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. The largest nonrepresentable integer $\max{\rm NR}$, the number of nonrepresentable positive integers $\sum_{n\in{\rm NR}}1$ and the sum of nonrepresentable positive integers $\sum_{n\in{\rm NR}}n$ have been widely studied for a long time as related to the famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas for the weighted sum $\sum_{n\in{\rm NR}}\lambda^{n}n^\mu$, where $\mu$ is a nonnegative integer and $\lambda$ is a complex number. We also examine power sums of nonrepresentable numbers and some formulae for three variables. Several examples illustrate and support our results.