Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Abstract

Iserles et al. (J. Approx. Theory 65:151-175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115-140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.

Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

“…In [1] it has been studied properties of the zeros of orthogonal polynomials with respect to Gegenbauer-Sobolev inner product where the associated pair of measures does not form a symmetrically coherent pair.…”

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

“…Some examples of symmetric measures whose sequences of orthogonal polynomials satisfy (10) have been studied in [28]. Asymptotic properties of the corresponding sequences of orthogonal polynomials and the location of their zeros were analyzed in [29,30] for the Gegenbauer case, as well as in [31,32] for the Hermite case. The aim of the present contribution is to find all the symmetric pairs of measures such that (10) holds.…”