Central and local limit theorems for the coefficients of polynomials of binomial type

Abstract

We introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials P n ( x) of binomial type; that is, a sequence P n ( x) satisfying exp(xg(u)) = ∑ n=0 ∞ P n(x)( u n n! ) for some (formal) power series g( u) lacking constant term. We give a complete answer in the case when g( u) is a polynomial, and point out the widest known class of nonpolynomial power series g( u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of P n ( x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g( u) which assures these conditions on the coefficients of P n ( x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of P n ( x) are developed.