New trigonometric identities and generalized Dedekind sums
The author obtains new kinds of reciprocity type identities involving cotangent and cosecant functions by using complex analysis, which are analogous to those for parametrized cotangent sums, and from those identities two theorems are derived. First, by taking limits in the identities, reciprocity laws similar to the one for classical Dedekind sums are obtained. Secondly, by comparing coefficients in Laurent series expansions of trigonometric functions appearing in the identities, relations between values of the Hurwitz zeta function at odd positive integers and Bernoulli numbers are obtained. Also, the author finds that a parametrized cotangent sum is a generating function for Apostol's generalized Dedekind sums \(s_{2n-1}(q, p)\), from which the injectivity of the map from \(q/p\text{\,mod\,}\mathbb{Z}\) in \(\mathbb{Q}/\mathbb{Z}\) to the sequence \(\{s_{2n-1}(q,p)\}_{n\in\mathbb{N}}\) is established.
scientific article; zbMATH DE number 2064092
scientific article