On limiting distributions of Graham, Knuth, Patashnik recurrences

Abstract

Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by $$\left|\genfrac{}{}{0ex}{}{n}{k}\right|=(\alpha n+\beta k+\gamma )\left|\genfrac{}{}{0ex}{}{n-1}{k}\right|+({\alpha }^{\prime }n+{\beta }^{\prime }k+{\gamma }^{\prime })\left|\genfrac{}{}{0ex}{}{n-1}{k-1}\right|+I_{n=k=0}$$ for $0\le k\le n$ and six parameters $\alpha ,\beta ,\gamma ,{\alpha }^{\prime }{\beta }^{\prime },{\gamma }^{\prime }$. 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 ${\alpha }^{\prime }=0$ and the remaining five parameters are non--negative.