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 )...

We prove pointwise bounds for two-parameter families of Jacobi polynomials. Our imply estimates a class functions arising from the spectral analysis distinguished Laplacians and sub-Laplacians on unit sphere in arbitrary dimension, are instrumental proof sharp multiplier theorems those operators.

We characterize the probability distributions of finite all order moments having generating functions for orthogonal polynomials of ultraspherical type. 1. Motivation: Meixner families There is a one to one correspondance between probability distributions on the real line and polynomials of a one variable satisfying a three-terms recurrence relation subject to some positive conditions ([9]). Th...