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 (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

${\displaystyle {n \choose k}={\underset {z}{\mathrm {res} }}\;{\frac {(1+z)^{n}}{z^{k+1}}}={\frac {1}{2\pi i}}\int _{|z|=\rho }{\frac {(1+z)^{n}}{z^{k+1}}}\;dz}$

where ${\displaystyle 0<\rho <\infty }$

• Second binomial coefficient integral

${\displaystyle {n \choose k}={\underset {z}{\mathrm {res} }}\;{\frac {1}{(1-z)^{k+1}z^{n-k+1}}}={\frac {1}{2\pi i}}\int _{|z|=\rho }{\frac {1}{(1-z)^{k+1}z^{n-k+1}}}\;dz}$

where ${\displaystyle 0<\rho <1}$

• Exponentiation integral

${\displaystyle n^{k}=k!\;{\underset {z}{\mathrm {res} }}\;{\frac {\exp(nz)}{z^{k+1}}}={\frac {k!}{2\pi i}}\int _{|z|=\rho }{\frac {\exp(nz)}{z^{k+1}}}\;dz:}$

where ${\displaystyle 0<\rho <\infty }$

• Iverson bracket

${\displaystyle [[k\leq n]]={\underset {z}{\mathrm {res} }}\;{\frac {z^{k}}{z^{n+1}}}{\frac {1}{1-z}}={\frac {1}{2\pi i}}\int _{|z|=\rho }{\frac {z^{k}}{z^{n+1}}}{\frac {1}{1-z}}\;dz}$

where ${\displaystyle 0<\rho <1}$

• Stirling number of the first kind

${\displaystyle \left[{n \atop k}\right]={\frac {n!}{k!}}\;{\underset {z}{\mathrm {res} }}\;{\frac {1}{z^{n+1}}}\left(\log {\frac {1}{1-z}}\right)^{k}={\frac {n!}{k!}}{\frac {1}{2\pi i}}\int _{|z|=\rho }{\frac {1}{z^{n+1}}}\left(\log {\frac {1}{1-z}}\right)^{k}\;dz}$

where ${\displaystyle 0<\rho <1}$

• Stirling number of the second kind

${\displaystyle \left\{{n \atop k}\right\}={\frac {n!}{k!}}\;{\underset {z}{\mathrm {res} }}\;{\frac {(\exp(z)-1)^{k}}{z^{n+1}}}={\frac {n!}{k!}}{\frac {1}{2\pi i}}\int _{|z|=\rho }{\frac {(\exp(z)-1)^{k}}{z^{n+1}}}\;dz}$

where ${\displaystyle 0<\rho <\infty .}$