Ein analytisches Reziprozitätsgesetz. (An analytic law of reciprocity) (Q1084562)

If \(1<m\in {\mathbb{Z}}\) and \(r\in \{1,2,...,m-1\}\), let \[ \Phi_{r,m}(z)=\sum^{\infty}_{k=0}z^{km+r}/(km+r)!=(z^ r/r!)\prod^{\infty}_{n=1}(1+z^ m/\nu_{r,m}(n)^ m) \] denote the values of the r-th hyperbolic function of order m, where \(0<\nu_{r,m}(1)<\nu_{r,m}(2)<\nu_{r,m}(3)<...\), and let \[ \zeta_{r,m}(s)=m^{s-1}\int^{\infty}_{0}(\Phi '_{r,m}(x)/\Phi_{r,m}(x)-1)x^{s-1} dx/\Gamma (s). \] Then the infinite product \[ \eta_{r,m}(t)=\exp (-\zeta_{r,m}(2)\sin (\pi /m)t/m\pi)\prod^{\infty}_{n=1}m \exp (- \nu_{r,m}(n)t)\Phi_{r,m}(\nu_{r,m}(n)t) \] may be regarded as a generalization of the Dedekind eta-function, to which it is reduced in the case of \(r=1\) and \(m=2\), because the functional equation (law of reciprocity) \[ \eta_{r,m}(1/t)=t^{r^ 2/m} \eta_{r,m}(t) \] is shown to be valid within the sectorial region, whre \(| \arg (t)| <\pi /m\) and \(\eta_{r,m}\) is a holomorphic function. The proof is given by means of the calculus of residues.