# Frobenius Coin-Exchange Generating Functions

## TL;DR

This paper explores generating functions related to the Frobenius coin-exchange problem, providing closed-form solutions for cases with two parameters and for integers with exactly k representations, advancing understanding of integer representability.

## Contribution

It derives closed-form generating functions for the Frobenius problem and for integers with exactly k representations, extending previous results and offering new formulas and corollaries.

## Key findings

- Closed-form generating function for two-parameter Frobenius problem
- Explicit formula for integers with exactly k representations
- Wide-ranging corollaries for integer representability

## Abstract

We study variants of the \emph{Frobenius coin-exchange problem}: given $n$ positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This problem and its siblings can be understood through generating functions with 0/1 coefficients according to whether or not an integer is representable. In the 2-parameter case, this generating function has an elegant closed form, from which many corollaries follow, including a formula for the Frobenius problem. We establish a similar closed form for the generating function indicating all integers with exactly $k$ representations, with similar wide-ranging corollaries.