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.