Log-convex solutions to the functional equation \(f(x+1)=g(x)f(x):\Gamma\)-type functions
Let \(g:(0,\infty)\to (0,\infty)\) be eventually log-concave and suppose that \(\lim_{x\to\infty} g(x+x_0)/g(x)=1\) for some \(x_0>0\). According to the theorem of W. Krull there exists exactly one eventually log-convex function \(\varphi:(0,\infty)\to (0,\infty)\) such that \((*)\) \(\varphi(x+1)= g(x)\varphi(x)\) for every \(x>0\) and \(\varphi(1)=1\). This function is called in the paper a \(\Gamma\)-type function, two examples of which are the gamma function \(\Gamma\) and the \(q\)-gamma function \(\Gamma_q\) generated, respectively, by \(g(x)=x\) and \(g(x)= (1-q^x)/(1-q)\) with \(0<q<1\). The author establishes for \(\Gamma\)-type functions analogues of Legendre's duplication formula, Gauss' multiplication formula, Stirling's formula, Euler's constant, and Weierstrass' infinite product for the gamma function. He uses also these functions to find log-convex solutions \(\varphi:(0,\infty)\to (0,\infty)\) of the equation \(\prod_{k=0}^{m-1} \varphi(x+\frac km)= h(x)\). \{Reviewer's remark: Geometrically convex solutions of \((*)\) are studied by \textit{D. Gronau} and \textit{J. Matkowski} [Result. Math. 26, No. 3-4, 290-297 (1994; Zbl 0876.39003)]\}.
scientific article; zbMATH DE number 1031625
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