The Method of Coefficients

can be found in closed form by means of certain transformations on generating func tions and the extraction of coefficients (see the end of section 3) suggested by Ira Gessel, and they comment: "He [Gessel] attributes this elegant technique, the 'method of coefficients,' to G. P. Egorychev." Actually, in his book Integral Representation and Computation of Combinatorial Sums, Egorychev [4] (see [3] for the Russian edition) deals with the representation of combinatorial expressions in terms of integrals and uses the notation restL(t) (in the Russian edition CoeftL(t)) for the residue of the for mal Laurent series L(t) (i.e., the coefficient of t~x in L(t) or in the present, widely accepted notation, [t~]]L(t)). The residue notation has been widely used because of the "change of variables" formula [t~l]g(t) = [t~l]g(f(t))f\t), where g(t) is a for mal Laurent series and f(t) is a formal power series having f(0) = 0 and f\0) ^ 0. It appears that this result is equivalent to the Lagrange inversion formula (see rule (K6) and section 5) but is more compact and easy to remember. The change of vari ables formula was first proved by Jacobi [14]; a discussion with application can be found in [7, p. 15]. Egorychev gives an equivalent formula [4, p. 16]. With the residue notation, the coefficient of tn in L(t) is written restt~n~lL(t), and this is equivalent to [t~l]t~n~lL(t) = [tn]L(t), which is certainly more direct. Egorychev's method is especially elegant when he does not complicate proofs with the use of integral representations. An interesting example [4, p. 28] is his derivation of the Grosswald identity,