Weighted Sylvester sums on the Frobenius set

TL;DR

This paper derives explicit formulas for weighted sums over nonrepresentable positive integers in terms of Apostol-Bernoulli numbers, extending understanding of Frobenius number-related sums.

Contribution

It provides explicit or Apostol-Bernoulli number-based formulas for weighted sums over Frobenius set elements, a novel approach in number theory.

Findings

Explicit formulas for weighted sums involving nonrepresentable integers.

Representation of sums in terms of Apostol-Bernoulli numbers.

Enhanced understanding of Frobenius number sums.

Abstract

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