An extension of Hölder's theorem on the Gamma function
Hölder's theorem states that Euler's Gamma function does not satisfy any nontrivial algebraic differential equation with coefficients from \(\mathbb C(z)\). In other words, \(\Gamma\) is differentially transcendental over \(\mathbb C(z)\) [\textit{O. Hölder}, Math. Ann. 28, 1--13 (1886; JFM 18.0440.02)]. The paper under review contains a generalization of the theorem. Let \(D\) be the ring of continuous real-valued functions on \(\mathbb R\) such that (1) \(D\supset\mathbb R(z)\), (2) for each \(f\in D\setminus\{0\}\) the zero set \(f^{-1}(0)\) is finite and the translation \(x\mapsto f(x+1)\) belongs to \(D\). Let \(I\) be an interval, \(I\subset(0,\infty)\). Denote by \(D_I\) the integral domain of restrictions \(f\,|\, I\) for \(f\in D\) and let \(D_I\langle\Gamma\rangle=D_I [\Gamma^{(n)}\,|\, I:n\in\mathbb N]\). It is proved that if 1) \(D_I\subset C^\infty(I)\), 2) \(D_I\) is closed under taking derivatives, 3) \(D_I\langle\Gamma\rangle\) is an integral domain then \(\Gamma\,|\, I\) is differentially transcendental over \(D_I\).
ring of continuous real-valued functions
scientific article; zbMATH DE number 1011397
scientific article