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).
where is a positive real less than the absolute value of the smallest nonzero singularity.
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
where
where
where
where in the first case and in the second
where in the first case and in the second.
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.
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 (observe that with the image of with 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 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 (and is contained in the image of ). 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.
We may use the change of variables rule 1.8 (5) from the Egorychev text (page 16) 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.
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