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

TL;DR

This paper analyzes a broad class of linear recurrence relations using bivariate exponential generating functions, providing a complete classification of their differential equations and solutions, with applications to combinatorial numbers.

Contribution

It offers a comprehensive classification and solution framework for a family of recurrence relations using bivariate generating functions, addressing a classical problem from Concrete Mathematics.

Findings

Complete classification of PDEs for the generating functions

Explicit solutions for all cases of the recurrence relations

Analysis of degeneracy in the combinatorial numbers

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.