Bivariate generating functions for a class of linear recurrences: General structure Articles uri icon

publication date

  • July 2014

start page

  • 146

end page

  • 165

issue

  • 1

volume

  • 125

international standard serial number (ISSN)

  • 0097-3165

electronic international standard serial number (EISSN)

  • 1096-0899

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.

keywords

  • Recurrence equations
    Exponential generating functions
    Row generating polynomials