Solve this functional equation:

$$\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$$

for $F(x)$ where $a > 0$ is a parameter. I know there is a trivial constant solution, $F(x) = 1$. Is there a non-constant solution?

Assume that $F(x)$ is continuous, and all the smoothness properties that you require to arrive to an answer. Note that I would accept the answer in the form of an infinite product or sum, or integral ....

I do not know if it has an analytical solution, and have no reason to expect that it does. It showed up in some calculations I was doing. But if there is a technique I can apply, it would be nice to know about it.

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    $\begingroup$ FWIW: There is at least one solution, $F(x)\equiv 1$. $\endgroup$ – AccidentalFourierTransform Aug 24 '17 at 18:21
  • $\begingroup$ @AccidentalFourierTransform Thanks, I knew that. Forgot to tell in the question I am looking for a non-trivial solution. $\endgroup$ – becko Aug 24 '17 at 18:33
  • $\begingroup$ Why the downvote? Leave a comment $\endgroup$ – becko Aug 25 '17 at 13:59
  • $\begingroup$ Oh, I got notified from that I comment and I thought you were referring to me. FWIW, the downvote is not mine. $\endgroup$ – AccidentalFourierTransform Aug 25 '17 at 14:06
  • $\begingroup$ Please, do not pose the question in a way it looks like an exercise. That will avoid downvotes. $\endgroup$ – Fernando Muro Aug 25 '17 at 22:09

(A very partial answer.) I would like to point out that the case $a=1$ is special. In this case, we immediately have $F(1)=1$ and $$F(x)=F\left(\frac{(1+x)^2}{4}\right).$$ Near the point $x=1$ there aren't any smooth solutions besides the trivial one.


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