New trigonometric identities and generalized Dedekind sums
type: Item
instance of: scholarly article
title: New trigonometric identities and generalized Dedekind sums
zbMATH Open document ID: 1048.11036
author: Shinji Fukuhara
published in: Tokyo Journal of Mathematics
publication date: 2004-05-18T00:00:00Z
review text: 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.
reviewed by: Kaori Ota
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 2064092
zbMATH Keywords: Dedekind sum
zbMATH Keywords: trigonometric identity
zbMATH Keywords: Hurwitz zeta function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208679
OpenAlex ID: W2059647931
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DOI: 10.3836/TJM/1244208679
description: scientific article; zbMATH DE number 2064092