I have the following equation on the functions $f_i$ for $i>1$, defined in $[0,1/4)$
$$f_i(x) = f_{i-1}(x) + \frac{x}{1-4x}f_{i-2}(x)$$
And assume that $f_0$ equals $c(x)$, the generating function of the Catalan numbers, and $f_1(x)=(c(x)-1)/x$.
Any idea on how can I get a closed form if it exists, or differential equation, or relation to well known polynomials?
Testing the form $f_i = g^i$, we get $g^2 = g + \frac{x}{1-4x} \implies g_{\pm} = \frac12 \pm \frac1{2\sqrt{1-4x}}$
So letting $f_i =a g_+^i + b g_-^i$, we get $c = a+b$ and $$\frac{c-1}x = \frac{a+b}2 + \frac{a-b}{2\sqrt{1-4x}} \implies a - b = \frac{\sqrt{1-4x}}x(2c - 2-x c) $$ From which we should get $$f_i = \frac12\left(c+ \frac{\sqrt{1-4x}}x(2x-2-xc) \right)\left(\frac12 + \frac1{2\sqrt{1-4x}} \right)^i + \frac12\left(c - \frac{\sqrt{1-4x}}x(2x-2-xc) \right)\left(\frac12 - \frac1{2\sqrt{1-4x}} \right)^i $$
Assuming the working's all right, does the form help?