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

$${\mathcal{P}}\left( {x,w} \right) = \sum\limits_{n = 0}^\infty {c_n P_n \left( x \right)w^n ,} $$

where {P_n}_n is the sequence of orthogonal polynomials with respect to u.