Generating functions and combinatorial identities
Wenchang Chu · University of Zagreb University Computing Centre (SRCE) · 1998
Computation of generating functions for renewal sequences is performed by means of the multivariate Lagrange expansion formulae due to Good (1960), which yields the multifold analogue of Carlitz' mixed generating function.As applications, the natural transition is demonstrated from Euler's binomial theorem and the classical Vandermonde convolution formula to Abel identities and Hagen-Rothe formulas, as well as their multifold analogues due to Mohanty & Handa (1969) and Carlitz (1977), respectively.For a complex parameter a and two complex functions A(x) and = B(x), consider the sequence {Cn ( a)} defined by generating = functionIn combinatorial computations and special functions, it is often necessary to determine the mixed generating function of the sequence {CII (a +bn) }.For this purpose, Carlitz [2, 1977] found a very useful formula and pursued its application to special functions.Recently, the author noticed that the famous Abel identities and the Hagen-Rothe identities are equivalent, respectively, to Euler's binomial theorem and Vandermonde's classical convolution formula when the mixed generating function of Carlitz is assumed as precondition.This fact will be illustrated in the first section.As natural generalization, the second and the third sections will deal with two kinds of multifold analogues and their applications to combinatorial identities of multivariate convolutions.Denote by C andNo, respectively, the sets of complex numbers and non-negative integers, with the n-fold tensor products C' andN'~in which the linear ordering "::::;:" is induced from the usual one.For two vectors i = (X1>X2,'" ,xn) E C' and fil = (ml,m2,'" ,mn) E NO, we formally define factorial product fil!= TI~=1mk!, coordinate sum Iii = EZ=1 Xb multivariate-monomial X" = TIZ=1x;*, scalar product (i, fil) = E~=l mkXb binomial product (~) = TI~=l (:~), and multinomial Mathematics subject classification (1991): OSA1S,OSAI9.