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I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with respect to composition, i.e. $f \circ g \neq g \circ f $ for all $n$.

I must admit that I don't have a very clear motivation for this question, other than curiosity and that it seemed like a challenging though interesting problem on its own. The sum may also go over $ -\infty$ to $\infty$.

If any research has been done on this problem already, I'd like to learn more about it.

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  • $\begingroup$ Are there any conditions on domain and range of $f$? If they are to be functions $\mathbb R\to\mathbb R$, then you can make $(f\circ g)(n)$ and $(g\circ f)(n)$ completely arbitrary: take some sequence $a_n$ of distinct reals which aren't integers, and define $f(n)=a_n,g(n)=a_n$, then define $f(a_n),g(a_n)$ in any way you want, e.g. to make the sums convergent and equal, and define $f,g$ on other reals arbitrarily. $\endgroup$
    – Wojowu
    Commented Jan 5, 2022 at 19:54
  • $\begingroup$ @Wojowu yeah I suspect requiring continuity of $f$ and $g$ on $\mathbb{R} \setminus A$ for some finite set $A$ could be a suitable condition. If that still leaves room open for trivial examples, please let me know $\endgroup$ Commented Jan 5, 2022 at 19:59
  • $\begingroup$ If you pick $a_n=n+1/2$, then you can still make the functions continuous with my method. $\endgroup$
    – Wojowu
    Commented Jan 5, 2022 at 20:03
  • $\begingroup$ @Wojowu Hm okay. I guess I should think it through somewhat more. I was thinking along the lines of $\sum_{n=-\infty}^{\infty} \operatorname{sinc}(n) = \sum_{n=-\infty}^{\infty} \operatorname{sinc}(n)^{2} = \pi $. These are 'natural' functions that don't require some careful construction. I'm sorry, I probably don't articulate it very well, but do you perhaps see what I'm trying to get at? (suppose $\sum_{n=-\infty}^{\infty} \operatorname{sinc}(n^{2}) = \sum_{n=-\infty}^{\infty} \operatorname{sinc}(n)^{2}$ would be true - which it isn't - then it would be a good example of the problem above) $\endgroup$ Commented Jan 5, 2022 at 20:20

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