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 $\alpha \ge 0$, put ${\xi }_0=-1$ and let ${\xi }_n$ be the largest zero of $R_n^{(\alpha +1,\alpha )}$ for $n\ge 1$. We prove that $$R_k^{(\alpha ,\alpha )}(x)=\underset{n\ge 0}{\min }{R}_n^{(\alpha ,\alpha )}(x),{\xi }_{k-1}\le x\le {\xi }_k,$$ for every $k\ge 1$. 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 $\alpha =0$.

arXiv commentNew Proposition 6.3 added