Limit laws for linear recurrences of Eulerian type

Abstract. We study linear recurrences of Eulerian type of the form

Pn(v) = (\alpha(v)n+\gamma(v))Pn-1(v) +\beta(v)(1-v)P'n-1(v) (n >= 1),

with P0(v) given, where \alpha(v), \beta(v) and \gamma(v) are in most cases polynomials of low degree. We characterize the various limit laws of the coefficients of Pn(v) for large n using the method of moments and analytic combinatorial toolsunder varying \alpha(v), \beta(v) and \gamma(v). We apply our results to more than two hundreds of concrete examples that we collected from the literature and from Sloane's OEIS. Not only most of the limit results are new, but they are unified in the same framework. The limit laws we worked out include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the richness and diversity of such a simple recurrence scheme, as well as the generality and power of the approaches used.