Egorychev method


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. 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. Should a series appear during summation that is not finite the contours must be chosen such as to make the series converge.
Some of the integrals employed by the Egorychev method are:
where
where
where
where
where
where

Example I

Suppose we seek to evaluate
which is claimed to be :
Introduce :
and :
This yields for the sum :
This is
Extracting the residue at we get
thus proving the claim. There are no convergence issues here as the
sums involved are finite and with and
not being negative we can choose any non-zero finite value for
and.

Example II

Suppose we seek to evaluate
Introduce
Observe that this is zero when so we may extend to
infinity to obtain for the sum
Now put so that
and furthermore
to get for the integral
This evaluates by inspection to
Here the mapping from to determines
the choice of square root. For the conditions on
and we have that for the series to converge we
require or or The closest that the image
contour of comes to the origin is
so we choose for example This also ensures that so does not intersect the branch
cut . For example
and will work.
This example also yields to simpler methods but was included here to demonstrate the effect of substituting into the variable of integration.

Computation using formal power series

We may use the change of variables rule 1.8 from the Egorychev text
on the integral :
with and We
get and find
with the inverse of.
This becomes
or alternatively
Observe that
so this is
and the rest of the computation continues as before.