Zeros of Jacobi and ultraspherical polynomials

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