Egorychev Method: A Hidden Treasure

Egorychev method is a potent technique for reducing combinatorial sums. In spite of the effectiveness of the method, it is not well known or widely disseminated. Our purpose in writing this manuscript is to bring light to this method. At the heart of this method is the representation of functions as series. The chief idea in Egorychev method is to reduce a combinatorial sum by recognizing some factors in it as coefficients in a series (possibly in the form of contour integrals), then identifying the parts that can be summed in closed form. Once the summation is gone, the rest can be evaluated via one of several techniques, which are namely: (I) Direct extraction of coefficients, after an inspection telling us it is the generating function (formal power series) of a known sequence, (II) Applying residue operators, and (III) Appealing to Cauchy’s residue theorem, when the coefficients alluded to appear as contour integrals. We present some background from the theory of complex variables and illustrate each technique with some examples. In concluding remarks, we compare Egorychev method to alternative methods, such as Wilf–Zeilberger theory, Zeilberger algorithm, and Almkvist–Zeilberger algorithm and to the performance of computer algebra systems.