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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 $P_n^{(α, β)}$ and $P_{n}^{(α,β+2)}$ 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 $P_n^{(α, β)}$ and $P_{n+1}^{(α,β)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $α$, $β$ with $α>-1$, $-2<β<-1$. The interlacing of zeros of $P_n^{(α,β)}$ and $P_m^{(α,β+t)}$ for $m,n\in\mathbb{N}$ is discussed for $α$ and $β$ in this range, $t\geq 1$, and new upper and lower bounds are derived for the zero of $P_n^{(α,β)}$ that is less than $-1$.