Asymptotics of a class of polynomial recurrences

We will consider sequences of polynomials defined by a recursion involving these polynomials and their first order derivatives. Recurrences like that are common in combinatorial probability and have been used and analyzed throughout the years. We will discuss a relatively recent progress on the subject and, in particular, new approaches that have been developed in the past few years and that allow treating such recurrences in in certain degree of generality.