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

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.

• FWIW: There is at least one solution, $F(x)\equiv 1$. – AccidentalFourierTransform Aug 24 '17 at 18:21
• @AccidentalFourierTransform Thanks, I knew that. Forgot to tell in the question I am looking for a non-trivial solution. – becko Aug 24 '17 at 18:33
(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.