Combinatorial Sums: Egorychev's Method of Coefficients and Riordan Arrays

G.P. Egorychev introduced a method which transforms combinatorial sums (e.g. sums involving binomial coefficients and also non-hypergeometric expressions arising in combinatorial context) into integrals. These integrals can be simplified using substitution or residue-calculus. With the help of this method one can compute combinatorial sums to which classical algorithms are not applicable. In this thesis we restrict to the residue functional instead of manipulating integral representations.We demonstrate among others how the Lagrange inversion rule can be applied to find closed forms for combinatorial sums. The special focus is laid on sums involving Stirling numbers and Bernoulli numbers that are not that easy to handle in comparison to sums over binomial coefficients. The latter sums can be handled e.g. with the application of Zeilberger’s algorithm. A related notion that will be discussed and used are Riordan arrays, a concept which we also use to handle non-trivial sums.