  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.
