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In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu' results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

Sums of products of two Gaussian q-binomial coeficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences. [ABSTRACT FROM AUTHOR]

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients. [ABSTRACT FROM AUTHOR]