On limiting distributions of Graham, Knuth, Patashnik recurrences

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