Request for interesting Functional Equations, of a specific type

Question

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?

Thanks!

https://math.stackexchange.com/questions/2346506/solving-for-multi-variable-functions ... https://math.stackexchange.com/questions/2362996/ffxfyfxy-fxy — Donald Splutterwit, Aug 10 '17 at 00:01
Oh & don't forget this one https://math.stackexchange.com/questions/2330402/solving-a-functional-equation-from-a-problem-solving-textbook ... lol $\ddot \smile$ — Donald Splutterwit, Aug 10 '17 at 00:03
Is this equation $0 = f(3x)f(x) - f(2x)^2 + f(x)^2$ your type? — Somos, Aug 10 '17 at 00:58
@Somos Wow, that's really hard. How do you suggest I try and solve the second one? — Franklin Pezzuti Dyer, Aug 12 '17 at 21:25
@Nilknarf It comes from sequence problems, $a_{n+1}=\sin(a_n)$, $a_{n+1}=\tan^{-1}(a_n)$ and so on. See my answer to question 2380638. — Somos, Aug 12 '17 at 21:32
@Somos I challenge you to produce a solution to $$f(x+1)=f(x)-f(x)^2$$ This problem has plagued my mind for a couple of days and I have begun to suspect that it has no solution. — Franklin Pezzuti Dyer, Aug 14 '17 at 23:23
@Nilknarf I solved this in 1999 via emails to David Rusin (was on the web ) $f(x)=1/(x+c-0+$ $\log(x+c-1/2+\log(x+c-17/24+\log(x+c-919/1152+\log(x+c-\dots)))))$ where $c$ is constant. If $c=0.767993786...$ then $f(1)=0.5$. — Somos, Aug 15 '17 at 00:57
@Somos Goodness. How do you come up with a solution like that? — Franklin Pezzuti Dyer, Sep 19 '17 at 23:41
Maybe this one : https://math.stackexchange.com/questions/4860595/functional-equations-f4x-f3x-fx-f2x-and-g4x-g3x-gx3-g ? — mick, Feb 10 '24 at 20:10

Somos · Accepted Answer · 2019-02-12T23:14:09.080

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.

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

Request for interesting Functional Equations, of a specific type

1 Answers1