Multiple cotangent and generalized eta functions (Q1591452)

The multiple cotangent function \(\text{Cot}_n(z,w)\) is defined as \[ {d\over dz}\bigl(-\zeta_n' (0,z,w)+(-1)^n \zeta_n'(0,w-z,w)\bigr) \] where \(w=(w_1,\dots,w_n)\) and \(\zeta_n'(s,z,w)\) is the derivative with respect to \(s\) of the multiple Hurwitz zeta function defined as the analytic continuation of the multiple series \(\sum_\Omega(z+\Omega)^{-s}\) with \(\Omega= \sum^n_{i=1} m_i w_i\). This generalizes the classic formula \[ \pi \text{cot} \pi z={d\over dz} \bigl(-\zeta'(0,z)-\zeta'(0,1-z)\bigr) \] for the ordinary cotangent. It is shown that the multiple cotangent shares many properties with the ordinary cotangent, such as periodicity, a multiplication formula, and a partial fraction decomposition. In the case \(n=2\) the double cotangent is expressed in terms of generalized eta functions treated by \textit{B. C. Berndt} [Trans. Am. Math. Soc. 178, 495--508 (1973; Zbl 0262.10015)] and \textit{J. Lewittes} [Trans. Am. Math Soc. 171, 469--490 (1972; Zbl 0253.10022)].