Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$?
$F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and for $a=\dfrac\pi2$ there are $F(x)=\sin x$ and $F(x)=\cos x$. But the three are connected by Euler's formula $e^{ix}=\cos x$ $+i\sin x$. Indeed, on a more general note, letting $F(x)=e^{\lambda x}$, we have $\lambda=\dfrac{W(-a)}{-a}$ where W is the Lambert W function. My question would be if these are the only ones, due to the special properties of the number e and the exponential function, or if there aren't by any chance more, which do not belong in the same family or category as these, i.e., which are not exponential or trigonometric in nature ? Thank you.
Look at these MO entries: https://mathoverflow.net/questions/114875/on-equation-fz1-fz-fz/114878#114878 , and https://mathoverflow.net/questions/156312/solve-fx-int-x-1x1-ft-mathrmdt/156315#156315 . They contain the answer to your question.
EDIT. To put it shortly, the answer is: "yes" and "no". In exactly the same sense as the answer to a simpler question: "Is every periodic function an exponential/trigonometric sum"? "Yes" for a physicist, and "no" for a mathematician. But every periodic function is a limit of exp/trig sums.