Zeros of Jacobi and ultraspherical polynomials

Suppose {P-n((alpha,beta))(x)}(n=0)(infinity) 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 is an element of 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 is an element of 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 <= t, s <= 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.