Jacobi 相位展开的超球背景，闭式表达

中文摘要 AI 辅助

$P_n^{(\alpha,\beta)}(\cos\theta)$ 的大阶相位在 $\Nn^{-1}=(n+\frac{\alpha+\beta+1}2)^{-1}$ 的每一阶上都携带一个加法常数 $\kappa_r$，该常数由相位方程无法确定。这些常数在 $(\alpha^2,\beta^2)$ 下反对称，并在超球对角线 $\alpha=\beta$ 上消失；在那里不消失的是振幅，我们将其完全计算出来：对角对数项的 Borel 变换为 \\[ \Psi_A(t)=\frac{e^{t}\bigl(\cosh2\alpha t-\cosh t\bigr)}{\sinh 2t}. \\] 证明是算术性的而非渐近性的，通过终止的半奇整数族和两个经典 Bessel 递推实现。其次，我们在此背景上进行横向线性化并解决横向问题：一个具有选票数分子和精确 Appell $F_3\to{}_2F_1$ 约化的物理曲线邻接性，使得横向切线成为零平衡 Gauss 函数的参数变化，在边界处分裂为 Gamma 闭式块，并通过求积法求解。由此，形式横向切线在所有阶上以闭式形式计算（其实部为 trigamma 级数，其虚部为显式 Genocchi--Bernoulli 中心差分求积），所需的扇形估计在专门附录中证明。第三，算术推论：Gamma 指数的 Genocchi 形式具有唯一的 $2$-进占优的全一划分，由此我们证明对所有 $m\ge0$ 有 $\nu_2(w_m)=-2\nu_2((2m)!)$：即相位常数的尖锐二进分母定律，$((2m)!)^2w_m\in\mathbb Z_2^\times$，在所有阶上成立。由 Wilf--Zeilberger 论证认证的共轭层分解、四个辅助赋值定律，以及一个应用于一般可接受实参数（$\alpha,\beta>-1$）的 Jacobi 零点计算，共同完善了整体图景。

英文摘要

The large-degree phase of $P_n^{(α,β)}(\cosθ)$ carries, at each order in $\Nn^{-1}=(n+\frac{α+β+1}2)^{-1}$, an additive constant $κ_r$ which the phase equation does not determine. These constants are antisymmetric in $(α^2,β^2)$ and vanish on the ultraspherical diagonal $α=β$; what does not vanish there is the amplitude, and we compute it completely: the Borel transform of the diagonal logarithm is \[ Ψ_A(t)=\frac{e^{t}\bigl(\cosh2αt-\cosh t\bigr)}{\sinh 2t}. \] The proof is arithmetic rather than asymptotic, through the terminating half-odd-integer family and two classical Bessel recurrences. Second, we linearise transversally over this background and resolve the transverse problem: a physical-curve contiguity with ballot-number numerators and an exact Appell $F_3\to{}_2F_1$ reduction render the transverse tangent a parameter variation of a zero-balanced Gauss function, split at the boundary into Gamma-closed blocks and solved by quadratures. The formal transverse tangent is thereby computed in closed form (trigamma series for its real part, an explicit Genocchi--Bernoulli central-difference quadrature for its imaginary part), in all orders, with the required sectorial estimates proved in a dedicated appendix. Third, the arithmetic consequence: the Genocchi form of the Gamma exponent has a unique $2$-adically dominant all-ones partition, from which we prove $ν_2(w_m)=-2ν_2((2m)!)$ for every $m\ge0$: the sharp dyadic denominator law for the phase constants, $((2m)!)^2w_m\in\mathbb Z_2^\times$, in all orders. A conjugate-layer factorisation certified by a Wilf--Zeilberger argument, four auxiliary valuation laws, and an application to the computation of Jacobi zeros for general admissible real parameters ($α,β>-1$) complete the picture.

发表机构

• IFCAE, Universidade de Vigo（维戈大学航空与空间工程学院）

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