Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

analysis functional-equations ordinary-differential-equations

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.

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