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Another characterization of the gamma function. (English) Zbl 1027.33001

Summary: The function \(\frac{\log\Gamma(x)}{\log x}\) is characterized to be the only convex solution of the functional equation \[ f(x+1)=\frac{\log x}{\log(x+1) }(f(x)+1), \quad x\in(0,\infty). \] Some relations to the function \({\log\Gamma(x+1)}/{x^a }\), \(0<a \leq 1\) are shown.

MSC:

33B15 Gamma, beta and polygamma functions
39A13 Difference equations, scaling (\(q\)-differences)
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