Zeros of Quasi-Orthogonal Jacobi Polynomials

TL;DR

This paper investigates the interlacing properties of zeros of Jacobi polynomials in quasi-orthogonal sequences with specific parameter ranges, providing necessary and sufficient conditions and new bounds for their zeros.

Contribution

It establishes conditions under which the interlacing conjecture by Askey holds for certain Jacobi polynomial zeros and proves non-interlacing results for specific cases.

Findings

Interlacing conditions depend on parameters nd eta.

Zeros of consecutive polynomials do not interlace for nd eta in the specified range.

New bounds for zeros less than -1 are derived.

Abstract

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 $-1$.