On the Taylor expansion of the Lerch zeta-function
The Lerch zeta function is the analytic continuation in the complex \(s\) plane of the series \[ L(x,a,s)=\sum_{n\geq 0}\exp\{2\pi inx\}(n+a)^{- s}, \] where \(x\) and \(a\) are real parameters. Properties of this function are deduced from its Taylor expansion in the parameter \(a\). By studying the function \(w(\xi)=L(x,\;a+\xi,\;s)\) in the neighborhood of \(\xi=0\) the author establishes a common source of various known summation formulas involving the Riemann zeta function \(\zeta(s)\) and the Hurwitz zeta function \(\zeta(s,a)\). The method also gives generalizations of some theorems that characterize these functions.
scientific article; zbMATH DE number 150668