Zeros of Quasi-Orthogonal Jacobi Polynomials

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha>-1$, $-2<\beta<-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(\alpha, \beta)}$ and $P_{n}^{(\alpha,\beta+2)}$ are interlacing, holds when the parameters $\alpha$ and $\beta$ are in the range $\alpha>-1$ and $-2<\beta<-1$. We prove that the zeros of $P_n^{(\alpha, \beta)}$ and $P_{n+1}^{(\alpha,\beta)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $\alpha$, $\beta$ with $\alpha>-1$, $-2<\beta<-1$. The interlacing of zeros of $P_n^{(\alpha,\beta)}$ and $P_m^{(\alpha,\beta+t)}$ for $m,n\in\mathbb{N}$ is discussed for $\alpha$ and $\beta$ in this range, $t\geq 1$, and new upper and lower bounds are derived for the zero of $P_n^{(\alpha,\beta)}$ that is less than $-1$.