Orthogonal polynomials on two or more intervals: the Sprugnoli context

Abstract

We show how the group of Sprugnoli arrays on three elements can be used to define families of polynomials that are orthogonal on two intervals, and we indicate how the group can be extended in the hierarchy of Sprugnoli groups to yield families of orthogonal polynomials with more complex supports. In particular, we examine the defining three term recurrences and the Jacobi matrices of these families of orthogonal polynomials, while detailing the Sprugnoli matrix structure of the coefficient matrices of the polynomial families. We also consider integral representations of the principal moments on the defining intervals. Sprugnoli arrays are shown to relate to periodic recurrences where the parameters remain cyclically constant.