Generating functions and companion symmetric linear functionals

Abstract

In this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that $$u\left({x 2n+1}}\right)=0$$. In some cases we can deduce explicitly the expression for the generating function $${\mathcal{P}}\left({x,w}\right)=\sum\limits_{n=0} infty {c_n P_n \left(x \right)w ,}$$ where {Pn}n is the sequence of orthogonal polynomials with respect to u.