The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers.
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 (usually this is done by integrating over a small circular contour enclosing the origin).
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
Suppose we seek to evaluate which is claimed to be :
{\displaystyle {k \choose j}={\frac {1}{2\pi i}}\int _{|w|=\gamma }{\frac {(1+w)^{k}}{w^{j+1}}}\;dw.}
This yields for the sum :
This is Extracting the residue at
we get thus proving the claim.
Suppose we seek to evaluate
Introduce Observe that this is zero when
to infinity to obtain for the sum Now put
small is another closed circle-like contour which makes one turn and which we may certainly deform to obtain another circle
) and furthermore to get for the integral This evaluates by inspection to (use the Newton binomial) Here the mapping from
determines the choice of square root.
For the conditions on
we have that for the series to converge we require
The closest that the image contour of
γ < ϵ −
does not intersect the branch cut
(and is contained in the image of
This example also yields to simpler methods but was included here to demonstrate the effect of substituting into the variable of integration.
We may use the change of variables rule 1.8 (5) from the Egorychev text (page 16) on the integral with
This becomes or alternatively Observe that
so this is and the rest of the computation continues as before.