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.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.2010
 arXiv:
 arXiv:1307.2010
 Bibcode:
 2013arXiv1307.2010B
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics
 EPrint:
 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