Bivariate Generating Functions for a Class of Linear Recurrences: General Structure

Abstract

We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many cases of combinatorial interest for particular choices of the six parameters that define it. We give a complete classification of the partial differential equations satisfied by the exponential generating functions, and solve them in all cases. We also show that the recurrence relations defining the combinatorial numbers appearing in this problem display an interesting degeneracy that we study in detail. Finally, we obtain for all cases the corresponding univariate row generating polynomials.

21 pages (LaTeX2e). Major changes with respect to version 1. Section 1 (resp. 2) is an abridged version of sections 1, 2, and 3 (resp. 4) of arXiv:1307.2010v1; section 3 contains new material, section 4 is an abridged version of section 3 of arXiv:1307.5624v1, and the appendix contains a summary of section 5 of arXiv:1307.2010v1

References in corpus (1)

• The method of characteristics, and "problem 89" of Graham, Knuth and Patashnik

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