Functions with periodic sequence of derivative-values

Question

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
E.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$.

Addendum:

As there doesn't seem to be an established name already, I'm tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another portmanteau.

These deriodic functions can be viewed as generalizing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function's value and the first $2n-1$ of its derivatives for a given $x_0$.

Answer 1

Assuming that $\mathbb N^+$ is defined as $\{1,2,\dots\}$, the set of functions in questions is just the set of all solutions $f$ of all the simple linear ODE's $f^{(k+1)}=f'$ with $k\in\mathbb N^+$, that is, the set of all functions $f$ such that $$f(x)=a+\sum_{j=0}^{k-1} c_{k,j}\exp\{e^{2\pi ij/k}x\}$$ for some $k\in\mathbb N^+$, some complex $a$ and $c_{k,j}$'s, and all $x$.

(Whether this set of functions has a name I do not know.)