Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros
Abstract
A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial Pn(α,β) contains a sequence of constants κ1,κ2,…, each of which is a polynomial in α2 and β2 with rational coefficients; for instance κ1=(α2−β2)/4. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of κr divides lcm(1,3,…,2r−1), and that the power of $2$ is at most 2Er with Er=3r−1+ν2((r−1)!), where ν2 is the $2$-adic valuation. The study of the leading coefficient leads to the formula ν2 (∑j=0mbinom mj1/(2j+1))=m+ν2(m+1) (m≥0), which we prove by showing that the series ∑k≥0k!/(2k+1)!!, whose real sum is π/2, converges to $0$ in the field ℚ2 of $2$-adic numbers. Finally, we show that the statement ``the exponent Er is attained for every r'' is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for r≤48.
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BibTeX
@misc{area2026dyadic,
title = {{Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros}},
author = {Iván Area},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23006},
eprint = {2608.23006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros
AU - Iván Area
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.23006
UR - https://arxiv.org/abs/2608.23006
AB - A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial P_n^(α,β) contains a sequence of constants κ_1,κ_2,…, each of which is a polynomial in α^2 and β^2 with rational coefficients; for instance κ_1=(α^2−β^2)/4. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of κ_r divides lcm(1,3,…,2r−1), and that the power of $2$ is at most 2^E_r with E_r=3r−1+ν_2((r−1)!), where ν_2 is the $2$-adic valuation. The study of the leading coefficient leads to the formula ν_2 (∑_j=0^mbinom mj1/(2j+1))=m+ν_2(m+1) (m≥0), which we prove by showing that the series ∑_k≥0k!/(2k+1)!!, whose real sum is π/2, converges to $0$ in the field ℚ_2 of $2$-adic numbers. Finally, we show that the statement ``the exponent E_r is attained for every r'' is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for r≤48.
ER -