Is there a closed form, or cleaner way of writing a function satisfying $\frac{d^nf(x)}{dx^n}|_{x=0}=f(n)$ for all $n$?

Question

Given the following, and assuming that $f(x)$ is infinitely differentiable: $$\frac{d^nf(x)}{dx^n}\Bigg|_{x=0}=f(n)$$ What functions $f$ could satisfy this equation? Do any functions of $f$ have a closed form, or if not does it have a form that is just a normal ODE form?

Answer 1

Let $f(x)=a^{x+1}$, where $a$ satisfies $\ln(a)=a$. Then $f^{(n)}(0)=\ln^n(a) a^{1}=a^{n+1}=f(n)$, as desired. Note that $a$ will be a complex number here, explicitly in terms of Lambert’s W: $a=e^{-W(-1)}\approx 0.318+1.337i$.