with *P*_{0}(*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 *P*_{n}(*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.