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The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients. The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem. The sought-for identity can now be found using manipulations of integrals. Some of these manipulations are not clear from the generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero, yielding a new expression for the original sum. The residue at infinity is particularly important in these considerations.Some of the integrals employed by the Egorychev method are:
where 0 < ρ < ∞ {\displaystyle 0<\rho <\infty }
where 0 < ρ < 1 {\displaystyle 0<\rho <1}
where 0 < ρ < ∞ . {\displaystyle 0<\rho <\infty.}