Zeta regularized product expressions for multiple trigonometric functions
Let \(m\) be a positive integer. The authors introduce a multiple analogue of the gamma function (different from the one defined by Barnes) defined by the zeta regularized product \[ G_m(x):=\exp(\phi'(0)), \] where \(\phi(s)\) is given by the Dirichlet series \[ \phi(s)=\sum_{n=1}^\infty\frac{n^m}{(n+x)^s}. \] The function \(G^{-1}_m(x)\) is an entire function of \(x\); \(G_m(0)=\exp(\zeta'(-m))\), where \(\zeta(s)\) is the Riemann zeta-function. Here \(G_m(x)\) is used to treat multiple trigonometric functions. Let \(\mathcal S_r(x)\) be the multiple sine function \[ \mathcal S_r(x)=e^\frac{x^{r-1}}{r-1}\prod_{n=-\infty, n\neq0}^\infty P_r\left(\frac x n\right)^{n^{r-1}}, \] where \[ P_r(u)=(1-u)\exp\left(u+\frac{u^2}2+\dots+\frac{u^r}2\right). \] The main result of the paper states, that for \(m\geq1\), \[ \mathcal S_{2m}(x)=G_{2m-1}(x)G^{-1}_{2m-1}(-x), \] \[ \mathcal S_{2m+1}(x)= \exp\left((-1)^m\frac{\zeta(2m+1)(2m)!}{2^{2m}\pi^{2m}}\right) G^{-1}_{2m}(x)G^{-1}_{2m}(-x). \] Similar formulas are derived for multiple cosine function. Also the duplication formula relating \(\mathcal S_m(2x)\) and \(\mathcal S_m(x)\) is obtained.
scientific article; zbMATH DE number 2145304