An alternating Mizuno formula, modified gamma functions, and double sine special values

Abstract

For $n\ge 1$ and $\mathrm{Re}(x_j)>0$, we prove an alternating analogue of Mizuno's product formula: \begin{equation*} \prod\nolimits_{m=0}^{\infty} \left(\prod_{j=1}^{n}(m+x_j)\right)^{(-1)^m} = \frac{(\sqrt{π/2})^n} {\prod_{j=1}^{n}\widetildeΓ(x_j)}, \end{equation*} where $\tilde{\Gamma }$ is the modified gamma function associated with the alternating Hurwitz zeta function. The polynomial factors are regularized using factorwise logarithms, and the formula establishes compatibility between this regularization and polynomial factorization. Applications include Gaussian and cyclotomic (Kurokawa--Wakayama type) evaluations; the single-factor case recovers the alternating Lerch formula, and a normalized ratio yields Wallis' product. A second group of results concerns an alternating double sine-type function. At nonexceptional positive periods, its positive-integer special values admit finite tangent-product representations. These products define locally meromorphic continuations $F_m$ in the period parameter. For $m\ge 2$ and nonzero algebraic parameters, the continued values belong to ${\mathbb{P}}^1(\overline{\mathbb{Q}})$ at rational parameters and are finite, nonzero, and transcendental at nonrational parameters. We determine their orders at rational parameters and distinguish parameter continuation from fixed-period specialization at exceptional parameters. Additional results include Barnes factorizations of multiple alternating gamma functions, a Glaisher--Kinkelin special value, finite-product Stirling-type asymptotics, a symmetric two-parameter form of the known odd-integer multiplication formula, and an odd-scaling distribution relation for the alternating double sine-type function.

arXiv comment23 pages