From geometry to Euler identities
The authors consider several elementary analytic transformations \(F\) of \({\mathbb R}^ 2\) which are connected to Euler identities and to a work of Ostrowski. Given \(M_ 0=(x_ 0,y_ 0)\) the sequence \(M_{n+1}=F(M_ n)\) converges in \({\mathbb R}^ 2\) to a limit \((x^{\infty},y^{\infty})\) and relations between \(x^{\infty}\) and \(y^{\infty}\), or properties of \(y^{\infty}\) happen to translate interesting arithmetical properties. In the first example, let \(M_ 0=(x_ 0,y_ 0)\) (with \(0<x_ 0<y_ 0)\), let \(P=(x_ 0,0)\), let \(Q=(0,y_ 0)\) and let \[ F(x,y)=(\frac{x^ 3}{x^ 2+y^ 2},\frac{y^ 3}{x^ 2+y^ 2}). \] Then \(M_ 1\) appears to be the orthogonal projection of \(M_ 0\) onto \(PQ\). The limit is \((0,y^{\infty})\) where \(y^{\infty}=\prod^{\infty}_{n=0}(1+x^{2\cdot 3^ n})^{-1}\) is the Cantor series and is transcendental for every algebraic \(x\) such that \(0<| x| <1.\) Other exciting studies are made with \[ F(x,y)=(\frac{x^ 2}{x^ 2+y^ 2},\frac{y^ 2}{x^ 2+y^ 2});\quad F(x,y)=(\frac{y^ 2}{x+y},- \frac{x(x+2y)}{x+y}) \] and so on.... The authors report that the examples were taken from a work of Ostrowski who studied the question of finding functions \(\phi(x)\) such that \(\prod^{\infty}_{n=0}(1+\phi(x)^ n)\) is an algebraic function of \(x\).
scientific article; zbMATH DE number 4116628