Zeros of Jacobi and Ultraspherical polynomials

TL;DR

This paper investigates the interlacing properties of zeros of Jacobi and ultraspherical polynomials, providing partial results, counterexamples, and extending known interlacing conditions for various parameter shifts.

Contribution

It offers new partial interlacing results for Jacobi and ultraspherical polynomials, clarifies when full interlacing does not hold, and includes numerical evidence to support these findings.

Findings

Zeros of certain Jacobi polynomial pairs are partially interlacing.

Full interlacing generally does not hold for shifted parameters.

Ultraspherical polynomial zeros are partially interlacing under specific conditions.

Abstract

Suppose $\{P_{n}^{(\alpha, \beta)}(x)\}_{n=0}^\infty $ is a sequence of Jacobi polynomials with $ \alpha, \beta >-1.$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $ P_{n}^{(\alpha,\beta)}(x)$ and $ P_{n+k}^{(\alpha + t, \beta + s )}(x)$ are interlacing if $s,t >0$ and $ k \in \mathbb{N}.$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $ P_{n}^{(\alpha,\beta)}(x)$ and $ P_{n+1}^{(\alpha, \beta + 1 )}(x),$ $ \alpha > -1, \beta > 0, $ $ n \in \mathbb{N},$ are partially, but in general not fully, interlacing depending on the values of $\alpha, \beta$ and $n.$ A similar result holds for the extent to which interlacing holds between the zeros of $ P_{n}^{(\alpha,\beta)}(x)$ and $ P_{n+1}^{(\alpha + 1, \beta + 1 )}(x),$ $ \alpha >-1, \beta > -1.$ It is known that the zeros of the equal degree Jacobi polynomials $ P_{n}^{(\alpha,\beta)}(x)$ and $ P_{n}^{(\alpha - t, \beta + s )}(x)$ are interlacing for $ \alpha -t > -1, \beta > -1, $ $0 \leq t,s \leq 2.$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $ P_{n}^{(\alpha,\beta)}(x)$ and $ P_{n}^{(\alpha + 1, \beta + 1 )}(x),$ when $ \alpha > -1, \beta > -1.$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $\alpha = \beta = \lambda -1/2$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $ C_{n}^{(\lambda)}(x)$ and $ C_{n + 1}^{(\lambda +1)}(x),$ $ \lambda > -1/2$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $ C_{n}^{(\lambda)}(x)$ and $ C_{n}^{(\lambda +3)}(x),$ $ \lambda > -1/2,$ is also discussed.