The Integral Representation Of An Algebraic Sum And Its Applications

In this paper, we firstly research the problem of the integral representation of the following algebraic sum by the Cauchy residue theorem in complex analysis. Where, m, n, s are nonnegative integers, and ai (i= 0,1,..., n) are not the same as each other. The next, we apply the results of these integral representations to study and give some calculating formulas of algebraic sum. As a result, many binomial summation formulas and some combinatorial identities are obtained. Such as, the results we got not only extended the consequences of J. Peterson, T. Mansour, Prodinger, P. Kirschenhofer, but also the different calculation formulas are given. At the same time, some new formulas and combinatorial identities are also obtained.