Some Partial Fraction Identities associated with the Cyclotomic Polynomials

TL;DR

This paper introduces the Extended Cover-Up Method for partial fraction decomposition of rational functions with cyclotomic polynomial denominators, enabling efficient computation of various number-theoretic functions and generalizing Fourier-Dedekind sums.

Contribution

It develops a novel algebraic approach for partial fractions, providing explicit formulas and a framework for generalizing Fourier-Dedekind sums and reciprocity theorems.

Findings

Derived direct formulas for q-partial fractions of generating functions.

Enabled pseudo-polynomial time computation of Sylvester denumerants, Frobenius number, and Ehrhart polynomials.

Unified explanation of reciprocity laws through Fourier analysis.

Abstract

We establish some partial fraction identities for rational functions whose denominators are implicit products of the cyclotomic polynomials. To achieve this, we first develop a general algebraic approach for partial fraction decomposition inspired by the Heaviside's cover-up method. We thus call our method the Extended Cover-Up Method. Using our method we obtain direct formulas for $q$-partial fractions for certain generating functions. As a direct consequence of our formulas one can compute the Sylvester denumerants, the Frobenius number and the Ehrhart polynomials in pseudo-polynomial time. Further, we provide a framework for a generalization of the Fourier-Dedekind sum and their associated Rademacher reciprocity theorem extending the results of Carlitz, Zagier and Gessel. By performing a Fourier analysis we demonstrate that our extended cover-up method explains in simple terms the mechanism behind the reciprocity law.