Duplication formulas in triple trigonometry
type: Item
instance of: scholarly article
title: Duplication formulas in triple trigonometry
zbMATH Open document ID: 1044.33010
author: Nobushige Kurokawa
author: Masato Wakayama
published in: Proceedings of the Japan Academy. Series A
publication date: 2004-07-12T00:00:00Z
review text: The multiple sine and cosine functions of degree \(r\geq2\) are respectively \[ S_r(x)=e^{x^r-1\over r-1}\prod_{n=1}^\infty \left\{P_r\left({x\over n}\right) P_r\left(-{x\over n}\right)^{(-1)^{r-1}}\right\}^{n^{r-1}} \] and \[ C_r(x)=\prod_{n=1}^\infty \left\{P_r\left({x\over n-{1\over2}}\right) P_r\left(-{x\over n-{1\over2}}\right)^{(-1)^{r-1}}\right\}^{(n-{1\over2})^{r-1}} \] where \(P_r(u)=(1-u)\exp\left(u+{u^2\over2}+\cdots+{u^r\over r}\right)\). These infinite products are generalisations of \(S_1(x)=2\sin(\pi x)= 2\pi x\prod_{n=1}^\infty\left(1-{x^2\over n^2}\right)\). The duplication formulae in question are the elementary formula \(S_r(2x)=S_r(x)^{2^{r-1}}C_r(x)^{2^{r-1}}\) and more complex formulae which rely on certain special values of the functions. For example, \(S_3({1\over2})=2^{1\over4}\exp\left({-7\zeta(3)\over8\pi^2}\right)\) from which can be derived \[ S_3(2x)=\exp\left({7\zeta(3)\over2\pi^2}\right) S_3(x)^4 S_3\left(x+{1\over2}\right)^4 S_2\left(x+{1\over2}\right)^{-4} S_1\left(x+{1\over2}\right). \] To see this, check that both sides are 1 at \(x=0\) and the logarithmic derivatives of both sides are \(8\pi x^2\cot(2\pi x)\). The authors also show in the same way that there are power series \(\Phi\) and \(\Psi\) such \(S_3(2x)=\Phi(S_3(x))\) around \(x=1\) and \(C_3(2x)^4=\Psi(C_3(x)^4)\) around \(x=1/2\).
reviewed by: John H. Loxton
Mathematics Subject Classification ID: 33E20
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 2078915
zbMATH Keywords: multiple sine and cosine functions
MaRDI profile type: Publication
full work available at URL: pjaa.79.123
OpenAlex ID: W1964445457
cites work: Multiple sine functions and Selberg zeta functions
cites work: Gamma factors and Plancherel measures
cites work: Q3137235
cites work: Multiple sine functions
cites work: Q4544426
cites work: On \(\zeta(3)\)
cites work: Higher Selberg zeta functions
cites work: EXTREMAL VALUES OF DOUBLE AND TRIPLE TRIGONOMETRIC FUNCTIONS
cites work: On q-Basic Multiple Gamma Functions
cites work: Q4838351
DOI: 10.3792/PJAA.79.123
description: scientific article; zbMATH DE number 2078915