Mizuno 乘积公式的交替推广与修正 gamma 函数
Alternating generalizations of Mizuno's product formula and the modified gamma functions
- South China University of Technology（华南理工大学）
- Kyungnam University（庆南大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文建立了 Mizuno 乘积公式的完整交替推广，引入修正 gamma 函数的交替版本，并推导出 Wallis 乘积、Kurokawa--Wakayama 型公式及多重交替 gamma 函数的性质，最后给出 Shintani 双正弦函数交替模拟的算术二分法。
AI中文摘要：
设 $\tilde{\Gamma}(x)$ 表示作者最近引入的修正 gamma 函数，它是经典欧拉 gamma 函数的交替模拟。本文建立了 Mizuno 著名乘积公式的完整交替推广：\begin{equation*} \prod_{m=0}^{\infty}\left(\prod_{j=1}^{n}(m+x_{j})^{(-1)^{m}}\right) =\frac{\left(\sqrt{\frac{\pi}{2}}\right)^n}{\prod_{j=1}^{n}\tilde{\Gamma}(x_{j})} =\prod_{j=1}^{n}\left(\prod_{m=0}^{\infty}(m+x_{j})^{(-1)^{m}}\right). \end{equation*} 该恒等式我们称之为交替 Mizuno 公式，它将经典 gamma 函数 $\Gamma$ 和常数 $\sqrt{2\pi}$ 替换为其自然的交替对应物 $\tilde{\Gamma}$ 和 $\sqrt{\pi/2}$。作为直接推论，我们恢复了交替 Lerch 公式，并且通过特化到 $x=2$ 并乘以 2，得到 Wallis 著名乘积 \begin{equation*} \frac{2\cdot2}{1\cdot3}\cdot\frac{4\cdot4}{3\cdot5}\cdot\frac{6\cdot6}{5\cdot7}\cdot\cdots=\frac{\pi}{2} \end{equation*} 的一个极其简洁的推导。更一般地，通过利用多项式因式分解，我们推导出分圆域上交替乘积的 Kurokawa--Wakayama 型公式。除了这些乘积恒等式之外，我们研究了多重交替 gamma 函数 $\bar\Gamma_N(x)$，获得了用 Barnes 多重 gamma 函数表示的显式因式分解、涉及 Glaisher--Kinkelin 常数的闭式求值，以及 Gauss--Legendre 型乘法公式。最后，我们引入了 Shintani 双正弦函数的交替模拟，并建立了一个清晰的算术二分法：对于代数数 $\tau$，其在整数点处的特殊值当且仅当 $\tau$ 为有理数时是代数的，否则是超越的。
英文摘要：
Let $\tildeΓ(x)$ denote the modified gamma function recently introduced by the authors as an alternating analogue of the classical Euler gamma function. In this paper, we establish a complete alternating generalization of Mizuno's celebrated product formula: \begin{equation*} \prod_{m=0}^{\infty}\left(\prod_{j=1}^{n}(m+x_{j})^{(-1)^{m}}\right) =\frac{\left(\sqrt{\fracπ{2}}\right)^n}{\prod_{j=1}^{n}\tildeΓ(x_{j})} =\prod_{j=1}^{n}\left(\prod_{m=0}^{\infty}(m+x_{j})^{(-1)^{m}}\right). \end{equation*} This identity, which we refer to as the alternating Mizuno formula, replaces the classical gamma function $Γ$ and the constant $\sqrt{2π}$ by their natural alternating counterparts $\tildeΓ$ and $\sqrt{π/2}$. As immediate consequences, we recover the alternating Lerch formula and, by specializing to $x=2$ and multiplying by $2$, a remarkably concise derivation of Wallis' famous product \begin{equation*} \frac{2\cdot2}{1\cdot3}\cdot\frac{4\cdot4}{3\cdot5}\cdot\frac{6\cdot6}{5\cdot7}\cdot\cdots=\fracπ{2}. \end{equation*} More generally, by exploiting polynomial factorizations, we derive Kurokawa--Wakayama type formulas for alternating products over cyclotomic fields. Beyond these product identities, we investigate the multiple alternating gamma functions $\barΓ_N(x)$, obtaining explicit factorizations in terms of Barnes' multiple gamma functions, closed-form evaluations involving the Glaisher--Kinkelin constant, and a Gauss--Legendre type multiplication formula. Finally, we introduce an alternating analogue of Shintani's double sine function and establish a clean arithmetic dichotomy: for algebraic $τ$, its special values at integer points are algebraic precisely when $τ$ is rational, and transcendental otherwise.
