Calculation of multiple combinatorial sums in the theory of holomorphic functions in Cn

At the end of the 1970ʼs, G.P. Egorychev developed a method of coefficients, which found successful applications for work with combinatorial sums. In this article, with the method of coefficients two identities were proved. One of the identities was proved in 2008, using the integral representation of holomorphic functions in domains of special form. Theorem 2 was proved with application of the coefficients method, it is a generalization of the results of Shelkovick and Zeilberger in the case of z1+⋯+zn=1.