On conjectures by Csordas, Charalambides and Waleffe

TL;DR

This paper proves new interlacing properties of zeros for special polynomials derived from Jacobi polynomials, extending previous results and confirming conjectures within broader parameter ranges.

Contribution

It establishes new interlacing results for polynomials from Jacobi polynomials, broadening the known parameter ranges and confirming conjectures.

Findings

Interlacing of polynomials from $P_n$ and $P_{n-1}$ holds in wider parameter ranges.

Interlacing between polynomials from $P_n$ and $P_{n-2}$ is also confirmed.

Results extend previous conjectures and provide new insights into polynomial zero distributions.

Abstract

In the present note we obtain new results on two conjectures by Csordas et al. regarding the interlacing property of zeros of special polynomials. These polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue problem. Their coefficients are the successive even derivatives of the Jacobi polynomials $P_n(x;\alpha,\beta)$ evaluated at the point one. The first conjecture states that the polynomials constructed from $P_n(x;\alpha,\beta)$ and $P_{n-1}(x;\alpha,\beta)$ are interlacing when $-1<\alpha<1$ and $-1<\beta$. We prove it in a range of parameters wider than that given earlier by Charalambides and Waleffe. We also show that within narrower bounds another conjecture holds. It asserts that the polynomials constructed from $P_n(x;\alpha,\beta)$ and $P_{n-2}(x;\alpha,\beta)$ are also interlacing.