Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform :
$$f(x) = \int_0^{\infty} g(x,t) \space dt $$
Or maybe $$f(x) = \int_0^{\infty} \int_0^{\infty} h(x,t,r) \space dt \space dr$$
How fast does $f(x)$ grow ?
Let $f(x)=a^{g(x)}$ , where $a\in\mathbb{R}^+$ and $a\neq1$ ,
Then $a^{g(x+1)}=(a^{g(x)})^{\ln(x)}$
$a^{g(x+1)}=a^{\ln(x)g(x)}$
$g(x+1)=\ln(x)g(x)$
$g(x)=\Theta(x)\prod\limits_x\ln(x)$, where $\Theta(x)$ is an arbitrary periodic function with unit period
$\therefore f(x)=a^{\Theta(x)\prod\limits_x\ln(x)}$ , where $\Theta(x)$ is an arbitrary periodic function with unit period, $a\in\mathbb{R}^+$ and $a\neq1$