I've been trying to find a function with the property $$f(x)=\frac{f(2x)}{x+1}$$ using elementary functions only, and it has proven to be harder than I thought.
Does anyone know how to go about finding a function given one of its properties in a systematic manner?
I can already guess that this function will use a logarithm (base 2) somewhere in it, but I still can't find the function.
Does it exist in the elementary functions? Does it exist elsewhere? Please help!
If we assume $f(0)=1$ we have that a solution of $$f(x)= \left(1+\frac{x}{2}\right)\cdot f\left(\frac{x}{2}\right) $$ over the interval $x\in(-2,2)$ is given by $$\begin{align} f(x)&=\prod_{n\geq 1}\left(1+\frac{x}{2^n}\right) \\ &=\exp\sum_{n\geq 1}\log\left(1+\frac{x}{2^n}\right) \\ &=\exp\sum_{n\geq 1}\sum_{m\geq 1}\frac{(-1)^{m+1}x^m}{m2^{mn}}, \end{align}$$ i.e., by $$ f(x)=\exp\sum_{m\geq 1}\frac{(-1)^{m+1}x^m}{m(2^m-1)}. $$