Partial-fraction decomposition of a rational function and its application

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