The Gamma function is a meromorphic function such that \begin{equation} \Gamma(z+1) = z\, \Gamma(z), \quad \Gamma(1) = 1. \end{equation} The Bohr–Mollerup theorem states that the Gamma function is the unique solution to this functional equation, under the condition that $ \log \Gamma(x) $ be convex on the positive reals.
Let $f$ be an entire function.
- Under what conditions on $f$ does there exist a meromorphic function $G$ such that $$ G(z+1) = f(z)\, G(z), \quad G(1) = 1? $$
- Under what conditions does a unique solution exist?
- Are there any such 'generalised Gamma functions' that have standardised names?
