A new Rational Generating Function for the Frobenius Coin Problem

TL;DR

This paper introduces a new rational generating function for the Frobenius Coin Problem, enabling efficient determination of whether a sum can be formed from given coin denominations.

Contribution

It develops a linear recurrence relation for the generating function, allowing it to be expressed as a rational function with bounded polynomial degrees.

Findings

The generating function G(x) is rational, expressed as P(x)/Q(x).

The recurrence relation for G(x) is linear.

Degrees of P(x) and Q(x) are bounded by the largest coin denomination.

Abstract

An important question arising from the Frobenius Coin Problem is to decide whether or not a given monetary sum S can be obtained from N coin denominations. We develop a new Generating Function G(x), where the coefficient of x^i is equal to the number of ways in which coins from the given denominations can be arranged as a stack whose total monetary worth is i. We show that the Recurrence Relation for obtaining G(x), is linear, enabling G(x) to be expressed as a rational function, that is, G(x) = P(x)/Q(x), where both P(x) and Q(x) are Polynomials whose degrees are bounded by the largest coin denomination.