发表机构
Universidade de Vigo（维戈大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究雅可比多项式零点相位常数的分母，证明其相关2-adic整性定律，将推测的精确分母定律转化为逐系数表述，且该定律已由配套论文证明。
AI 中文摘要
雅可比多项式零点的渐近相位包含未被相位方程确定的加性常数κᵣ。我们将它们的分母作为A=α²和B=β²的多项式进行研究，证明denκᵣ的奇数部分整除最小公倍数lcm(1,3,…,2r-1)，且2^{Eᵣ}κᵣ是2-adic整数，其中Eᵣ=3r-1+ν₂((r-1)!)。极值系数由赋值定律ν₂(∑_{j=0}^{m}C(m,j)1/(2j+1))=m+ν₂(m+1)控制，该定律源于ℚ₂中的恒等式∑_{k≥0}k!/(2k+1)!!=0。我们还将推测的精确分母定律转化为逐系数表述：若Φ是勒让德切线的博雷尔变换且W(t)=sinh(2t)ImΦ(t)/t²=∑_{m≥0}wₘt^{2m}，则精确定律等价于对每个m，((2m)!)²wₘ∈ℤ₂^×且保持各指标相等。该最终整性表述（下文猜想W）已在本系列的配套论文中证明，故精确分母定律在所有阶成立；本文建立了正规形和赋值精确转移，并记录了界定该证明的确切证据与结构障碍。
英文摘要
A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial $P_n^{(α,β)}$ contains a sequence of constants $κ_1,κ_2,\dots$, 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 $\operatorname{lcm}(1,3,\ldots,2r-1)$, 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 \[ ν_2\!\left(\sum_{j=0}^{m}\binom mj\frac1{2j+1}\right)=m+ν_2(m+1)\qquad(m\ge0), \] which we prove by showing that the series $\sum_{k\ge0}k!/(2k+1)!!$, whose real sum is $π/2$, converges to $0$ in the field $\mathbb Q_2$ 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 $r\le48$.
