An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

TL;DR

This paper introduces an algebraic method for deriving explicit formulas for Sylvester denumerants using q-partial fractions, involving Bernoulli numbers and generalized Fourier-Dedekind sums, advancing the understanding of their structure.

Contribution

The work develops an algebraic approach to Sylvester's denumerants, providing new explicit formulas and structural insights using q-partial fractions and generalized sums.

Findings

Derived new explicit formulas for Sylvester waves

Proved reciprocity theorems for generalized Fourier-Dedekind sums

Established a structure theorem for top-order terms of the denumerants

Abstract

In 1857 Sylvester established an elegant theory that certain counting functions (which he termed denumerants) are quasi-polynomials by decomposing them into periodic and non-periodic parts. Each component of the decomposition, called a Sylvester wave, corresponds to a root of unity. Recently several researchers, using either combinatorial arguments or complex analytic techniques, obtained explicit formulas for the waves. In this work, we develop an algebraic approach to the Sylvester's theory. Our methodology essentially relies on deriving $q$-partial fractions of the generating functions of the denumerants, and thereby obtain new explicit formulas for the waves. The formulas we obtain are expressed in terms of the degenerate Bernoulli numbers and a generalization of the Fourier-Dedekind sum. Further, we also prove certain reciprocity theorems of the generalized Fourier-Dedekind sums and a structure result on the top-order terms of the waves. The proofs rely on our evaluation operator and our far-reaching generalization of the Heaviside's cover-up method for partial fractions. SageMath code for this work is available in the public domain.