An asymptotic distribution theory for Eulerian recurrences with applications

Abstract

We study linear recurrences of Eulerian type of the form \[ P_n(v) = (\alpha(v)n+\gamma(v))P_{n-1}(v) +\beta(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where $\alpha(v), \beta(v)$ and $\gamma(v)$ are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of $P_n(v)$ for large $n$ using the method of moments and analytic combinatorial tools under varying $\alpha(v), \beta(v)$ and $\gamma(v)$, and apply our results to more than two hundred of concrete examples when $\beta(v)\ne0$ and more than three hundred when $\beta(v)=0$ that we gathered from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.