Zeros of Quasi-Orthogonal Jacobi Polynomials

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.