On limiting distributions of Graham, Knuth, Patashnik recurrences
Pawel Hitczenko
TL;DR
This work analyzes the limiting distributions of integer-valued random variables naturally associated with Graham–Knuth–Patashnik recurrences when $\alpha'=0$ and the remaining parameters are nonnegative. Using the bivariate exponential generating function $F(z,x)$, derived via a PDE and solved by characteristics in the $\alpha'=0$ case, the authors apply algebraic-singularity schema to establish central limit theorems for $X_n$ and, in boundary regimes (notably $\alpha=\beta=0$), identify degenerate or alternative limit laws such as negative-binomial and Poisson-type limits. The main contributions include explicit formulas for the limiting mean and variance (often linear in $n$) across several parameter regimes, a detailed treatment of the $\beta'>0$ and $\beta'=0$ cases, and a clear delineation of the boundary cases where normal limits fail and other distributions arise. Collectively, the results deepen the probabilistic understanding of GKP recurrences and illustrate the power of the PDE/characteristics approach combined with algebraic singularity methods in deriving limiting distributions for combinatorial 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 $$\left|{ n\atop k}\right|=(αn+βk+γ)\left|{n-1\atop k}\right|+(α' n+β' k+γ')\left|{n-1\atop k-1}\right|+I_{n=k=0}$$ for $0\le k\le n$ and six parameters $α,β,γ,α'β',γ'$. 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 $α'=0$ and the remaining five parameters are non--negative.