Asymptotic normality of the coefficients of polynomials related to the classical system orthogonal ones

Abstract

Suitable transformations on the classical system of orthogonal polynomials lead to polynomials with nonnegative coefficients. Asymptotic normality of those coefficients is established by local limit theorem. Applications include, important combinatorial numbers, as the number of connections of a telephone exchange, the coefficients of a rook polynomial, the coefficients of the card matching polynomial and others. Asymptotic formulas for such numbers are also provided whenever it is possible.