Infinite Matrices in the Theory of Orthogonal Polynomials

Abstract

In this chapter we present a brief account of a matrix approach to the study of sequences of orthogonal polynomials. We use an algebra of infinite matrices of generalized Hessenberg type to represent polynomial sequences and linear maps on the complex vector space of all polynomials. We show how the matrices are used to characterize and to construct several sets of orthogonal polynomials with respect to some linear functional on the space of polynomials. The matrices allow us to study several kinds of generalized difference and differential equations, to obtain explicit formulas for the orthogonal polynomial sequences with respect to given bases, and also to obtain formulas for the coefficients of the three-term recurrence relations. We also construct a family of hypergeometric orthogonal polynomials that contains all the families in the Askey scheme and a family of basic hypergeometric q-orthogonal polynomials that contains all the families in the q-Askey scheme.