Zeros of Jacobi and Ultraspherical polynomials

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 $\alpha -t>-1,\beta >-1,$ $0\le t,s\le 2.$ 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.

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