An extension of Hölder's theorem on the Gamma function
type: Item
instance of: scholarly article
title: An extension of Hölder's theorem on the Gamma function
zbMATH Open document ID: 0901.46023
DOI: 10.1007/BF02774035
author: Chris Miller
published in: Israel Journal of Mathematics
publication date: 1998-09-07T00:00:00Z
review text: 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 \(D_I\).
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 1011397
zbMATH Keywords: differential ring
zbMATH Keywords: quasi-analytic
zbMATH Keywords: Hölder's theorem
zbMATH Keywords: Euler's gamma function
zbMATH Keywords: differentially transcendental
zbMATH Keywords: ring of continuous real-valued functions
zbMATH Keywords: integral domain
zbMATH Keywords: JFM 18.0440.02
MaRDI profile type: Publication
Wikidata QID: Q56812628
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description: scientific article; zbMATH DE number 1011397