Zeros of Quasi-Orthogonal Jacobi Polynomials

Abstract

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by α>−1, −2<β<−1. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials P(α,β)n and P(α,β+2)n are interlacing, holds when the parameters α and β are in the range α>−1 and −2<β<−1. We prove that the zeros of P(α,β)n and P(α,β)n₊₁ do not interlace for any n∈N, n≥2 and any fixed α, β with α>−1, −2<β<−1. The interlacing of zeros of P(α,β)n and P(α,β+t)m for m,n∈N is discussed for α and β in this range, t≥1, and new upper and lower bounds are derived for the zero of P(α,β)n that is less than −1.