On a characterization theorem of E. Artin
The following Artin type characterizations of \(\Gamma:\mathbb{R}_+\to\mathbb{R}_+\) are proved. Assume that \(f:\mathbb{R}_+\to\mathbb{R}_+\) satisfies the Gauss multiplication formula \((M_p)\) for some fixed \(p\geq 2\). Then \(f=\Gamma\) in either of the following cases: a) \(f\) absolutely continuous on \([\alpha,\alpha+1]\) for some \(\alpha\in(0,1]\) and \(\lim_{x\to0} xf(x)=1\), b) \(f\) absolutely continuous on \([\alpha,\alpha+1]\) for some \(\alpha\in (0,1]\), \(\lim_{x\to0} xf(x)\in [0,\infty]\) and \(f(1)=1\), c) \(f\) absolutely continuous on \([{1\over 2},1+\varepsilon]\) for some \(\varepsilon>0\), \(f(1)=1\) and \(f\) satisfies the reflection formula. The optimality of these characterizations is checked by means of counterexamples. We state one: For any given \(m\in\mathbb{N}\) and \(p\geq 2\) there is a function \(F:\mathbb{R}_+\to\mathbb{R}_+\) of class \(C^m(\mathbb{R}_+)\) satisfying \((M_p)\), \(F(1)=1\) and \(F\neq\Gamma\).
characterizations of the gamma function by means of functional equations
scientific article; zbMATH DE number 878371