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Can I solve functional equations with Mathematica? How?

For example, I am trying to solve:

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

for specific values of $a$. Can Mathematica help me? I would be happy with a numerical solution.

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    $\begingroup$ As far as I've been able to tell, you solve these kinds of equations in Mathematica the same way you would do them by hand: by finding some transformation that turns the equation into a recurrence relation or a differential equation or a algebraic equation, and solve that instead. There are various "tricks" you can find online just by searching "solve functional equations". $\endgroup$ – march Aug 25 '17 at 21:08
  • $\begingroup$ @march I have tried looking but didn't find any. $\endgroup$ – becko Aug 26 '17 at 6:33
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    $\begingroup$ The equation yields4 F[1/a] == 1, if that is of help. $\endgroup$ – bbgodfrey Aug 27 '17 at 8:09

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