Stirling modular forms and special values of multiple cotangent functions (Q693019)

Let \(r\) be a positive integer, and let \(D_r=\{(\omega_1,\cdots,\omega_r)\in (\mathbb{C}^{*})^r\}\), where \(\omega_1,\cdots,\omega_r\) are belonging to one side with respect to a line crossing \(0\). For \(\mathbf{\omega}=(\omega_1,\cdots,\omega_r)\in D_r\), the multiple sine functions and cotangent functions are defined by \[ S_r(x,\mathbf{\omega})=\coprod_{n_1,\cdots,n_r\geq 0}(n_1\omega_1+\cdots+n_r\omega_r+x)\times \left(\coprod_{n_1,\cdots,n_r\geq 1}(n_1\omega_1+\cdots+n_r\omega_r-x)\right)^{(-1)^{r-1}} \] and \[ \text{Cot}_{r}(x,\mathbf{\omega})=\frac{d}{dx}\log S_r(x,\mathbf{\omega}), \] where \(\coprod\) denotes the regularized product \[ \coprod_{\lambda\in \Lambda}\lambda=\exp\left(-\frac{d}{ds}\sum_{\lambda\in \Lambda}\lambda^{-s}\Big{|}_{s=0}\right). \] The Stirling modular form \(\rho_r\) is defined as \[ \rho_r(\mathbf{\omega})=\mathop{\coprod_{n_1,\cdots,n_r\geq 0}}_{(n_1,\cdots,n_r)\neq (0,\cdots,0)}(n_1\omega_1+\cdots+n_r\omega_r). \] This paper studied special values of multiple cotangent functions, and obtained some relations between special values of multiple cotangent functions. Furthermore, a new proof of the modularity of the Dedekind eta function was given by using a transformation of Stirling modular forms.