Some characterizations of the Euler gamma function
Some characterizations of the Euler gamma function
Assume that $f:(0,\infty)\rightarrow (0,\infty ) $ is bounded from above on a set of positive Lebesgue measures or on a set of the second category with the Baire property and satisfies the functional equation $f(x+1)=xf(x)$ for $x>0$ and $f(1)=1$. We prove that, if there is a positive sequence $(p_{n})$, $\lim_{n\rightarrow …