Complete Bernstein functions and scaled ultraspherical zeros
Abstract
Let zn,j(λ) denote the positive zeros, in decreasing order, of the ultraspherical polynomial Cnλ, λ>−1/2, with the reduced limiting interpretation at λ=0 specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint λ=0 is singular. For every n≥3, √(λ+1) zn,j(λ) becomes a complete Bernstein function after translation of its parameter interval to (0,∞). For every n≥2, √λ zn,j(λ) is a complete Bernstein function on (0,∞). For every n≥4, the largest-zero trajectory √(λ+(2n2+1)/(4n+2)) zn,1(λ) has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling λzn,j(λ).
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BibTeX
@misc{castillo2026complete,
title = {{Complete Bernstein functions and scaled ultraspherical zeros}},
author = {K. Castillo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19186},
eprint = {2609.19186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19186},
note = {CC0},
}
RIS
TY - UNPB
TI - Complete Bernstein functions and scaled ultraspherical zeros
AU - K. Castillo
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.19186
UR - https://arxiv.org/abs/2609.19186
AB - Let z_n,j(λ) denote the positive zeros, in decreasing order, of the ultraspherical polynomial C_n^λ, λ>−1/2, with the reduced limiting interpretation at λ=0 specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint λ=0 is singular. For every n≥3, √(λ+1) z_n,j(λ) becomes a complete Bernstein function after translation of its parameter interval to (0,∞). For every n≥2, √λ z_n,j(λ) is a complete Bernstein function on (0,∞). For every n≥4, the largest-zero trajectory √(λ+(2n^2+1)/(4n+2)) z_n,1(λ) has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling λz_n,j(λ).
ER -