A proof of Gautschi's conjecture on subrange Jacobi polynomials
Abstract
Let πn be the monic polynomial of degree n orthogonal on [−c,c], 0<c≤1, with respect to the Jacobi weight (1−x)α(1+x)β, where −1<α<β. Gautschi conjectured that [ (πn(−c))/πn(c) ]2 ((1−c)/(1+c))β−α <1. By his variation formula, this inequality is sufficient for every positive zero of πn to move to the right as c increases. For 0<c<1, a first-crossing argument proves the conjecture throughout 0<α<β. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles β≥0. In the negative wedge, writing α=−r−λ and β=−r+λ, an ensemble Ward bound yields the region c2≤3/(3+r). A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every n≥1, −1<α<β, and 0<c≤1. We also give a direct degree-one proof and an explicit asymptotic limit. The case c=1 is immediate.
Related
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BibTeX
@misc{botta2026proof,
title = {{A proof of Gautschi's conjecture on subrange Jacobi polynomials}},
author = {V. Botta and K. Castillo and L. Tertuliano da Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.05963},
eprint = {2608.05963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05963},
note = {CC0},
}
RIS
TY - UNPB
TI - A proof of Gautschi's conjecture on subrange Jacobi polynomials
AU - V. Botta
AU - K. Castillo
AU - L. Tertuliano da Silva
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.05963
UR - https://arxiv.org/abs/2608.05963
AB - Let π_n be the monic polynomial of degree n orthogonal on [−c,c], 0<c≤1, with respect to the Jacobi weight (1−x)^α(1+x)^β, where −1<α<β. Gautschi conjectured that [ (π_n(−c))/π_n(c) ]^2 ((1−c)/(1+c))^β−α <1. By his variation formula, this inequality is sufficient for every positive zero of π_n to move to the right as c increases. For 0<c<1, a first-crossing argument proves the conjecture throughout 0<α<β. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles β≥0. In the negative wedge, writing α=−r−λ and β=−r+λ, an ensemble Ward bound yields the region c^2≤3/(3+r). A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every n≥1, −1<α<β, and 0<c≤1. We also give a direct degree-one proof and an explicit asymptotic limit. The case c=1 is immediate.
ER -