Year: 2026
Type: preprint
Abstract: Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $α\ge0$, put $ξ_0=-1$ and let $ξ_n$ be the largest zero of $R_n^{(α+1,α)}$ for $n\ge1$. We prove that $$ R_k^{(α,α)}(x) =\min_{n\ge0}R_n^{(α,α)}(x), \quad ξ_{k-1}\le x\leξ_k, $$ 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 $α>0$ and $-1
Source: arXiv (Cornell University)
Authors: Castillo, K., Sadigova, S.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G5712257541, G8985288032
OpenAlex ID: W7213356360