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arXiv · 2006.12548

Some Partial Fraction Identities associated with the Cyclotomic Polynomials

N. Uday Kiran
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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.

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N. Uday Kiran. 2020-06-04. Some Partial Fraction Identities associated with the Cyclotomic Polynomials. https://arxiv.org/abs/2006.12548

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