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Let $\tilde\Gamma(z)$ be the modified gamma function introduced by theauthors in a recent preprint "arXiv2106.14674". In this note, we obtain thefollowing 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-\zetay)} \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' 1656fomula \begin{equation*}\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot5}\frac{6\cdot6}{5\cdot 7}\cdots=\frac{\pi}{2}. \end{equation*}

Mizuno-type result and Wallis' formula