PreprintCC0arXiv
The lower envelope of ultraspherical polynomials
Abstract
Let Rn(a,b)=Pn(a,b)/Pn(a,b)(1) be the normalised Jacobi polynomials. For α≥0, put ξ0=−1 and let ξn be the largest zero of Rn(α+1,α) for n≥1. We prove that Rk(α,α)(x) =minn≥0Rn(α,α)(x), ξk−1≤ x≤ξk, for every k≥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 α>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 α=0.
Related
BibTeX
@misc{castillo2026lower,
title = {{The lower envelope of ultraspherical polynomials}},
author = {K. Castillo and S. Sadigova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15473},
eprint = {2609.15473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15473},
note = {CC0},
}
RIS
TY - UNPB
TI - The lower envelope of ultraspherical polynomials
AU - K. Castillo
AU - S. Sadigova
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.15473
UR - https://arxiv.org/abs/2609.15473
AB - Let R_n^(a,b)=P_n^(a,b)/P_n^(a,b)(1) be the normalised Jacobi polynomials. For α≥0, put ξ_0=−1 and let ξ_n be the largest zero of R_n^(α+1,α) for n≥1. We prove that R_k^(α,α)(x) =min_n≥0R_n^(α,α)(x), ξ_k−1≤ x≤ξ_k, for every k≥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 α>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 α=0.
ER -