Mizuno-type result and Wallis' formula

TL;DR

This paper derives new infinite product formulas involving a modified gamma function, generalizing Wallis' formula and establishing connections to Kurokawa–Wakayama and Lerch-type formulas.

Contribution

It introduces Mizuno-type results and related infinite product formulas involving the modified gamma function, extending classical identities like Wallis' formula.

Findings

Derived Mizuno-type infinite product formula involving the modified gamma function

Established a Kurokawa–Wakayama type formula for products with polynomial factors

Recovered Wallis' formula as a special case when setting specific parameters

Abstract

Let $\tilde\Gamma(z)$ be the modified gamma function introduced by the authors in a recent preprint "arXiv2106.14674". In this note, we obtain the following Mizuno-type result: \begin{equation*} \prod_{m=0}^{\infty}\left\{\prod_{j=1}^{n}(m+z_{j})\right\}^{(-1)^{m}}=\frac{\left(\sqrt{\frac{\pi}{2}}\right)^n}{\prod_{j=1}^{n}\tilde\Gamma(z_{j})}, \end{equation*} which imply a Kurokawa--Wakayama type formula \begin{equation*} \prod_{m=0}^\infty\left((m+x)^{n}-y^n\right)^{(-1)^{m}} =\frac{\left(\sqrt{\frac{\pi}{2}}\right)^n}{\prod_{\zeta^{n}=1}\tilde\Gamma(x-\zeta y)} \end{equation*} and a Lerch-type formula \begin{equation*} \prod_{m=0}^\infty(m+x)^{(-1)^{m}}=\frac{\sqrt{\frac{\pi}{2}}}{\tilde\Gamma(x)}. \end{equation*} By setting $x=1$ in the above result, we recover Wallis' 1656 fomula \begin{equation*}\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\frac{\pi}{2}. \end{equation*}