Egorychev method


The Egorychev method is a collection of techniques 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.
The main integrals employed by the Egorychev method are:
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.

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. This example also yields to simpler methods but was included here to demonstrate the effect of substituting into the variable of integration.