Evaluation formulas of Cauchy-Mellin type for certain series involving hyperbolic functions (Q2910121)

For \(\tau\in\mathbb{C}\) with \(\mathrm{Im}(\tau)>0\), define NEWLINE\[NEWLINE S(s;\tau) :=\sum^{\infty}_{m=1}\frac{(-1)^m}{\sinh(m\pi i/\tau)m^s}, \qquad s\in\mathbb{C}. NEWLINE\]NEWLINE The special value \(S(4k-1;i)\) (\(k\in\mathbb{Z}\)) was studied by Cauchy. In this paper, using the theory of the Eisenstein series and the Barnes multiple zeta-functions, the authors evaluate the values \(S(-4k+1;i)\) and \(S(-6k+1;\rho)\) (\(k\in\mathbb{N}\)) with \(\rho=e^{2\pi i/3}\); these can be respectively expressed as rational multiples of \((\varpi/\pi)^{4k}\) and \((\widetilde{\varpi}/\pi)^{6k}\) where \(\varpi:=\Gamma(1/4)^2/(2\sqrt{2\pi})\) is the lemniscate constant and \(\widetilde{\varpi}:=\Gamma(1/3)^3/(2^{4/3}\pi)\).NEWLINENEWLINEMoreover, the authors consider the following double analogue \(S_2(s;\tau)\) of \(S(s;\tau)\) defined by NEWLINE\[NEWLINE S_2(s;\tau) :=\sum_{m\in\mathbb{Z}\setminus\{0\}}\sum_{n\in\mathbb{Z}\setminus\{0\} \atop m+n>0}\frac{(-1)^{m+n}}{\sinh(m\pi i/\tau)\sinh(n\pi i/\tau)(m+n)^s}, \qquad s\in\mathbb{C}. NEWLINE\]NEWLINE It is then proved that \(S_2(-4k;i)\) and \(S_2(4k;i)\) (\(k\in\mathbb{N}\)) are expressed as a multiple of \(S(-4k+1;i)\) and a linear combination of \(S(4k+1;i)\) and \(S(4k-4j-1;i)\) for \(0\leq j\leq k\), respectively.