Contour Integral And Combinatorial Identities

Combinatorial identities is one of the most important contents in combinatorial mathematics.The methods of investigate combinatorial identities have combinatori-al method,number theory method,probabilistic method,special function method and complex analysis.In this paper,we will study several kinds of combinatorial identities by means of the method of complex analysis,and some applications will be presented.The paper is organized as following three parts.1.Norlurd obtained a contour representation of the following binomial sum in his classic book(also called Rice's integral):where C is a positively oriented closed curve and encloses 0,1,...,n,f(z)is an analytic function,g(z):=n!/z(z-1)…(z-n)is called kernel function.By constructing kernel function-s and integral contour,we generalize the Rice's integral into three kinds of form and their applications are given respectively.Especially,we get some combinatorial identities involving high order reciprocal of binomial coefficients.2.By using contour integral and Cauchy's residue theorem,via construct kernel functions and integral contour,we obtain the following symmetrical algebraic identity:where Bl(x1,..,xl)is Bell polynomials,xi,yi are determined parameters.We proved several main results that presented in Chu[J.Combin.Math.Combin.Comput.,60(2007),139-153]and Shen[Maste's degree thesis of Chongqing normal University,2016]by using the method of complex analysis.Furthermore,the result in Chu[Filomat,24(2010),41-46]as well as Xu and Cen[Ramanujan J.,40(2016),103-114]is simple corol-lary of our results.Additionally,we deduce numerous combinatorial identities involving Bell polynomials and harmonic numbers.In particular,the problem that can't be solved by partial fractional decomposition method is solved(Theorem 4.2).3.As the end of the paper,we give an integral representation of three kinds of binomial sums respectively.Additionally,we can deduce different kinds of combinatorial identities from those integrals.