Weighted Sylvester sums on the Frobenius set in more variables

TL;DR

This paper derives formulas for weighted sums involving nonrepresentable integers in the Frobenius problem using Eulerian numbers, extending to multiple variables and providing concrete examples.

Contribution

It introduces new formulas for weighted sums of nonrepresentable numbers in the Frobenius set, utilizing Eulerian numbers, and extends results to three variables.

Findings

Formulas for weighted sums involving nonrepresentable integers.

Extension of formulas to three-variable Frobenius problems.

Examples illustrating the derived formulas.

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.