On limiting distributions of Graham, Knuth, Patashnik recurrences

TL;DR

This paper investigates the limiting distributions of integer-valued random variables associated with a broad class of linear recurrences, providing a complete characterization under specific parameter conditions.

Contribution

It offers a comprehensive analysis of the asymptotic behavior of solutions to Graham, Knuth, and Patashnik recurrences when \\alpha'=0 and other parameters are non-negative.

Findings

Complete description of limiting distributions under specified parameters.

Identification of conditions leading to different distribution types.

Extension of previous recurrence solution analyses to probabilistic context.

Abstract

Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by $$\left|{ n\atop k}\right|=(\alpha n+\beta k+\gamma)\left|{n-1\atop k}\right|+(\alpha' n+\beta' k+\gamma')\left|{n-1\atop k-1}\right|+I_{n=k=0}$$ for $0\le k\le n$ and six parameters $\alpha,\beta,\gamma,\alpha'\beta',\gamma'$. Since then, a number of authors investigated various properties of the solutions of these recurrences. In this note we consider a probabilistic aspect, namely we consider the limiting distributions of sequences of integer valued random variables naturally associated with the solutions of such recurrences. We will give a complete description of the limiting behavior when $\alpha'=0$ and the remaining five parameters are non--negative.