I am looking for interesting functional equations of a specific type, and I thought that perhaps the math SE community would be able to deliver a good amount of them.

When I look up "functional equation problems", I usually get problems like $$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$$ and they usually have rather boring answers with solutions that are linear, constant, or nonexistent. The type of functional equation that I am looking for has only one variable (namely $x$) and often has a very strange answer using identities of various types of functions. For example, one of the easier equations is $$\alpha(x)+\alpha(2x)=1$$ I'm looking for non-boring (and thus non-constant) solutions, and so one solution to this equation is $$\alpha(x)=\sin^2(2\pi\log_2 x)$$ two examples of more complicated problems are $$\beta(x)+\beta\bigg(\frac{x-1}{x+1}\bigg)=\sin x$$ and $$\gamma(4000-400x)+\gamma(400-40x)+\gamma(40-4x)=x^2+x+1$$ The first has a very long solution, and the second has a polynomial solution... but I will exclude the solutions to these two and let you try them for yourselves, if you like.

Can anybody provide some examples of functional equations like this?



I have been doing research with elliptic and related functions and the functional equations they satisfy. What I usually am looking for is multivariable equations. As a simple example: $\sin(x+y)\sin(x-y)=\sin(x)^2-\sin(y)^2$ which is a solution of functional equation $0=f(x-y)f(x+y)-f(x)^2+f(y)^2$ which also has as solutions $f(x)=cx$.

I have many similar examples in Special Algebraic Identities which has many algebraic identities and some of them have interesting solutions when regarded as a functional equation. So the previous algebraic identity appears as $$\texttt{ id2_3_1_2a = +a*a -b*b -(a-b)*(a+b)}$$ with a $\texttt{[TS]}$ tag. What I don't have listed is a lot more identities with only a single variable because they make uninteresting identities in general. For example, for the $\sin(x)$ alone there are an unlimited number of single variable identites. I want to focus attention on the simplest of such identities. Here is a list of simple functional equations in one variable with interesting solutions:

$$ 0 = f(3x)^2 - f(5x)f(x) + f(3x)f(x) - f(x)^2 \tag{1}$$ $$ 0 = 4f(2x)^2 - 3f(3x)f(x) - f(x)^2 \tag{2}$$ $$ 0 = f(3x)f(2x) - 3f(3x)f(x) + f(2x)f(x) + f(x)^2 \tag{3}$$ $$ 0 = f(4x)f(2x)^2 - f(4x)f(3x)f(x) - f(3x)f(2x)f(x) + f(2x)f(x)^2 \tag{4}$$ $$ 0 = f(5x)f(x)^3 - f(4x)f(2x)^3 + f(3x)^3f(x) \tag{5}$$ $$ 0 = f(5x)(f(4x)-f(3x)-f(2x)+f(x)) - f(4x)(f(3x)+2f(x)) +\\ f(3x)(3f(2x)+3f(x)) +f(x)(f(x)-4f(2x)) \tag{6}$$

The last functional equation is the most challenging.

  • $\begingroup$ Aw, I can't get to your link. But still, very cool answer! I'll give them all a try! :D $\endgroup$ – Franklin Pezzuti Dyer Aug 11 '17 at 21:51
  • $\begingroup$ Goodness, these are hard! Here's one for you: $$g(2x)=4^x g(x)^2 g(x+1)^2$$ $\endgroup$ – Franklin Pezzuti Dyer Aug 12 '17 at 21:24

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