Alternating generalizations of Mizuno's product formula via modified gamma functions

Abstract

Let $\tilde{\Gamma}(x)$ 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{\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*} This identity, which we refer to as the alternating Mizuno formula, replaces the classical gamma function $\Gamma$ and the constant $\sqrt{2\pi}$ by their natural alternating counterparts $\tilde{\Gamma}$ and $\sqrt{\pi/2}$. As immediate consequences, we recover the alternating Lerch formula and, by specializing to $x=1$, 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=\frac{\pi}{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 $\bar\Gamma_N(x)$, 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 $\tau$, its special values at integer points are algebraic precisely when $\tau$ is rational, and transcendental otherwise.