**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 "**deri**vative-peri**odic**" or *deriodic* for short by means of inventing yet another [portmanteau](https://en.wikipedia.org/wiki/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: $n$ *deriodic*s suffice to approximate a function's value and the first $2n-1$ of its derivatives for a given $x\_0$.



Functions with periodic sequence of derivative-values
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.)


