The dyadic denominator law for the phase constants of the Jacobi zeros

Abstract

The asymptotic phase for the zeros of a Jacobi polynomial contains additive constants $\kappa_r$ that are not determined by the phase equation. We study their denominators as polynomials in $A=\alpha^2$ and $B=\beta^2$. We prove that the odd part of $\den\kappa_r$ divides $\operatorname{lcm}(1,3,\ldots,2r-1)$ and that $2^{E_r}\kappa_r$ is $2$-adically integral, where $E_r=3r-1+\nu_2((r-1)!)$. The extremal coefficient is governed by the valuation law \[ \nu_2\!\left(\sum_{j=0}^{m}\binom mj\frac1{2j+1}\right) =m+\nu_2(m+1), \] which follows from the identity $\sum_{k\ge0}k!/(2k+1)!!=0$ in $\mathbb Q_2$. We also transform the conjectural sharp denominator law into a single coefficientwise statement. If $\Phi$ is the Borel transform of the Legendre tangent and $W(t)=\sinh(2t)\operatorname{Im}\Phi(t)/t^2=\sum_{m\ge0}w_mt^{2m}$, then the sharp law is equivalent, with equality preserved at each index, to $((2m)!)^2w_m\in\mathbb Z_2^\times$ for every $m$. This final integrality statement (Conjecture~W below) has since been proved in the companion paper of this series, so the sharp denominator law holds in all orders; the present paper establishes the normal form and the valuation-exact transfer, and records the exact evidence and the structural obstructions that delimited that proof.