Integral Representation and Algorithms for Closed Form Summation

Abstract

The problem of computation and estimation of finite and infinite sums (generating functions) often arises in combinatorics and graph theory, theory of algorithms and computer algebra, group theory and function theory, probability theory and asymptotical analysis as well as in physics, statistical mechanics, and other areas of knowledge. This article is intended for a wide audience including graduate students and researchers in the various applied fields. Here we present the history, main results, model examples, various applications and perspectives of investigation in two connected general approaches to summation: the integral representation and computation of combinatorial sums (“the method of coefficients”) and the modern algorithmic approach. Each of the approaches is based on classic results of mathematical analysis, function theory and computer algebra.