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Egorychev method

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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:

  • First binomial coefficient integral

where

  • Second binomial coefficient integral

where

where

where

where

where

Example I

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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

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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.

Computation using formal power series

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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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References

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  • Egorychev, G. P. (1984). Integral representation and the Computation of Combinatorial sums. American Mathematical Society. ISBN 9780821898093.
  • Riedel, Marko; Mahmoud, Hosam. "Egorychev Method: A Hidden Treasure". La Matematica. doi:10.1007/s44007-023-00065-y.