I know the equation
$$ f^{\prime} (x) = f(x) f^{\prime} (x-1) $$
is solved by $f(x) = C$ or by tetration ( $ f(x+1) = \exp(f(x)) $).
So I wonder What are the solutions to
$$g^{\prime \prime}(x) = g(x) g^{\prime \prime} (x-1) ?$$
For $x>1$ and $g^{\prime \prime}(x) > 0$ ??
Any good asymptotics , functional equations , asymptotic functional equations or ... ???