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Can we find a function $f:\mathbb{R}\to\mathbb{R}$ , such that it's derivatives repeat after a certain period $T$. For example, exponential function $f(x)=Ae^x$ satisfies $f(x)=f'(x)=f''(x)=\cdots$ and in this case we can call the period to be $1$.

Now, I also found a function such that $f(x) \ne f'(x) $ but $f(x)=f''(x)$ and it repeats at a period of $2$. The example is $f(x)=Ae^x+Be^{-x}$.

However, I was unable to find a function whose period is $3$. Although, for period $4$, we have trigonometric functions, but I think it would be possible to find a class of functions, in which we can find such functions according to our wish by just varying $T$. It seems we can manipulate exponential functions so that it is possible for any $T$.

Any ideas would be appreciated!

I found this too: Functions that are their Own nth Derivatives for Real $n$ but I'm not sure if this is what I am looking for, because I am looking for real valued functions, and the answer given here gives functions in terms of $n$th roots of unity.

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This is just a basic ODE question. So we want to solve $$ y'''-y=0. $$ In an ODE class you will learn that we first have to solve the characteristic polynomial $$ x^3 -1 = 0. $$ I am sure you can see how it corresponds to our differential equation. A real solution would be $1$, and there are two complex and none-real solutions $z_1$ and $z_2 = \overline{z_1}$. Theory (which is not that hard, but not trivial either) then tells us, that all linear combinations of the functions $$ f_1(t) = \exp(t), ~f_2(t) = \exp(\mathrm{Re}(z_1)t) \cos(\mathrm{Im}(z_1)t), ~f_3(t) = \exp(\mathrm{Re}(z_1)t) \sin(\mathrm{Im}(z_1)t) $$ and ONLY those solve your problem.

For any $T$, you can proceed the same way. You will have to solve $$ x^T-1 = 0 $$ in $\mathbb{C}$ and you will get a solution $1$ and $T-1$ others. If $-1$ is among the latter (i.e. iff $T$ is even), you add $$ t \mapsto \exp(t), ~t \mapsto \exp(-t) $$ to the mix, and if not, for every complex (and non-real) pair $z, \bar z$ of roots you add $$ t \mapsto \exp(\mathrm{Re}(z)t) \cos(\mathrm{Im}(z)t), ~t \mapsto \exp(\mathrm{Re}(z)t) \sin(\mathrm{Im}(z)t). $$ Compute for yourself that it works.

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A concrete example of a real function with period 3 is the following: $$ f(x) := e^x+e^{-x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)+e^{-x/2}\cos\left(\frac{\sqrt{3}}{2}x\right). $$ This works because $f$ solves the differential equation $y'''=y$. However, as Meowdog points out, there is an entire class of functions that solve your problem.

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