The lower envelope of ultraspherical polynomials

Abstract

Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha\ge0$, put $\xi_0=-1$ and let $\xi_n$ be the largest zero of $R_n^{(\alpha+1,\alpha)}$ for $n\ge1$. We prove that $$ R_k^{(\alpha,\alpha)}(x) =\min_{n\ge0}R_n^{(\alpha,\alpha)}(x), \quad \xi_{k-1}\le x\le\xi_k, $$ for every $k\ge1$. This determines the pointwise lower envelope of the normalised ultraspherical polynomials and answers Question 2 posed by F. M. de Oliveira Filho in his 2009 thesis. For $\alpha>0$ and $-1<x<1$, we also determine all minimising degrees. The sequence is non-increasing up to each such degree, which answers his Question 1 in this range. A Legendre example gives a negative answer when $\alpha=0$.