Explicit evaluations and reciprocity theorems for finite trigonometric sums
type: Item
instance of: scholarly article
title: Explicit evaluations and reciprocity theorems for finite trigonometric sums
zbMATH Open document ID: 1011.11057
DOI: 10.1016/S0196-8858(02)00020-9
author: Bruce C. Berndt
author: Boon Pin Yeap
published in: Advances in Applied Mathematics
publication date: 2003-04-03T00:00:00Z
review text: For given integers \(m\geq 0\), \(h>0\), \(k>0\), and given real parameters \(a,b\) in the interval \([0,1)\), the authors consider trigonometric sums of the form \[ s_m(h,k;a,b)= \sum\cot^m \biggl(\pi \frac{r+b}{k} \biggr) \cot\Biggl(\pi \biggl( \frac{h(r+b)}{k}-a \biggr)\Biggr), \] where the sum is extended over \(0<r+b<k\). Using residue calculus they obtain a reciprocity law expressing \(s_m(h,k;a,b)/k+ s_m(k,h;b,a)/h\) in closed form in terms of polynomials evaluated at \(\cot(\pi a)\) and \(\cot(\pi b)\). Special cases include classical reciprocity laws for Dedekind sums and some of their generalizations, as well as some explicit evaluations of trigonometric sums. The paper also gives a valuable history and an extensive bibliography of evaluations and reciprocity theorems for trigonometric sums.
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 1892331
zbMATH Keywords: trigonometric sums
zbMATH Keywords: residue calculus
zbMATH Keywords: reciprocity law
zbMATH Keywords: Dedekind sums
zbMATH Keywords: extensive bibliography
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
cites work: Asymptotic Formulas and Generalized Dedekind Sums
cites work: Theorems on generalized Dedekind sums
cites work: Another Elementary Proof of Euler's Formula for ζ(2n)
cites work: The Reciprocity Law for Dedekind Sums via the Constant Ehrhart Coefficient
cites work: Dedekind cotangent sums
cites work: Elementary Evaluation of ζ(2n)
cites work: Dedekind Sums and a Paper of G. H. Hardy
cites work: Reciprocity theorems for Dedekind sums and generalizations
cites work: Analytic Properties of Arithmetic Sums Arising in the Theory of the Classical Theta-Functions
cites work: Beispiele zur Differentialtopologie von Singularitäten
cites work: Q4333506
cites work: Some integer-valued trigonometric sums
cites work: Some Diophantine relations involving circular functions of rational angles
cites work: A note on generalized Dedekind sums
cites work: Q3208731
cites work: Partial fractions and trigonometric identities
cites work: Finite cotangent sums and the Riemann zeta function
cites work: Cotangent sums, a further generlization of Dedekind sums
cites work: Heat kernel expansion on a generalized cone
cites work: Q4106142
cites work: Modular forms, generalized Dedekind symbols and period polynomials
cites work: New trigonometric identities and generalized Dedekind sums
cites work: Dedekind symbols associated with J-forms and their reciprocity law.
cites work: The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws
cites work: Q4108388
cites work: Some trigonometric identities encountered by McCoy and Orrick
cites work: Some trigonometric identities. II
cites work: Generating functions and generalized Dedekind sums.
cites work: Multiple Franel integrals
cites work: Dedekind-Rademacher Sums
cites work: Q4057485
cites work: Q5577712
cites work: Q5513537
cites work: Q5665935
cites work: Q4775913
cites work: Q5611941
cites work: A proof of a transformation formula in the theory of partitions
cites work: Q4733912
cites work: Multiple cotangent and generalized eta functions
cites work: Q3284008
cites work: Proof of a Conjecture of 1881 on Permanents
cites work: A note on trigonometric sums arising in gauge theory
cites work: Analyticity and integrability in the chiral Potts model.
cites work: Q4883480
cites work: Über eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung
cites work: On certain sums generating the Dedekind sums and their reciprocity laws
cites work: On a conjecture of R. F. Scott (1881)
cites work: Q5806065
cites work: Q4001741
cites work: On an extension of a theorem of S. Chowla
cites work: A Simple Proof of the Formula ∑ ∞ k = 1 = π 2 /6
cites work: Q3744481
cites work: Q4753775
cites work: Q4068685
cites work: Q5663963
cites work: Q5645846
cites work: Invariants of finite groups and their applications to combinatorics
cites work: On a trigonometrical sum
cites work: Q3901719
cites work: Conformal field theory and the cohomology of the moduli space of stable bundles
cites work: Exponential Sums of Lerch's Zeta Functions

Explicit evaluations and reciprocity theorems for finite trigonometric sums (cites work)
cites work: Q5524370

Explicit evaluations and reciprocity theorems for finite trigonometric sums (cites work)
cites work: On ∑∞n = 1 (1/n2k)

Explicit evaluations and reciprocity theorems for finite trigonometric sums (cites work)
cites work: Q4327101

Explicit evaluations and reciprocity theorems for finite trigonometric sums (cites work)
cites work: Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the G-signature theorem to transformation groups, symmetric products and number theory

Explicit evaluations and reciprocity theorems for finite trigonometric sums (cites work)
cites work: Higher dimensional Dedekind sums