according to question Finding solutions to $f'(x) = f(x + k)$

i ask generalization of this question

i am trying to find non-trivial functions $f \colon \mathbb R \to \mathbb R$ that $f^{n}(x) = f(x+k)$ with $k \in \mathbb R$,and $f^{n}$ is n'th derivate $f$

For $k<1$, I've found functions $f(x)= a^x$ that $a>1$,of course for large $n$ and some $a$ ,this is hold for $k\ge 1$

However, for $k>-1$,and $n$ be even I can only find a solution $f(x) = a^{-x}$, that $a>1$ .of course

for large $n$ and some $a$ ,this is hold for $k\le {-1}$

is there any other solution for values of $k$ and $n$?