Application Of The Residue Method In Proving Combinatorial Identities

In this paper, with the help of residue method we establish some identities involving the binomial coefficient, the unsigned Stirling numbers of the first kind, the Stirling num-bers of the second kind, the n-ordered Bell numbers, the Bernoulli polynomials of the first kind, the ordinary Bell polynomials, the Riemann Zeta Function, q-series, the unilateral hypergeometric series, the bilateral hypergeometric series and Γ-function. Showing that residue method is of great importance in proving combinatorial identities.Chapter 2:We evaluate the contour integral of auxiliary function with the Cauchy residue theorem. Then we give some summation formulas on the unilateral hypergeometric series ι+1Fι, the unilateral hypergeometric series 3F3, the bilateral hypergeometric series 2H1 and Γ-function.Chapter 3: By means of the residue method we obtain some q-series identities.Chapter 4:In this chapter, we introduce the concept of formal residue and several properties of formal residue at first. Using residue method some interesting formulas involving some special combinatorial sequences (the binomial coefficient, the unsigned Stirling numbers of the first kind, the Stirling numbers of the second kind, the n-ordered Bell numbers, the Bernoulli polynomials of the first kind, the ordinary Bell polynomials) come up accurately.