Duplication formulas in triple trigonometry
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 \(x=1/2\).
scientific article; zbMATH DE number 2078915