How can we pick two non-zero functions $f$ and $g$ so that for all integer $k\ge 0$

$$\sum_{n=1}^\infty f(n)g(n)^k = \left(\sum_{n=1}^\infty f(n)g(n) \right)^k$$

Assume $f$, $g$, and $k$ allow for convergence. Let neither $f$ nor $g$ involve $k$.

Note that $k=0$ implies $$\sum_{n=1}^\infty f(n) = 1$$

As @Kavi pointed out, solutions for $f$ are trivial when $g=1$. I would ideally wish for a general set of solutions, but I am definitely interested in any interesting not-obvious solutions.

Note: I made quite a few edits to simplify my question.

  • $\begingroup$ Can you specify any further about $k$? It seems like $f=h=0$ will do the job, depending on your restrictions for $k.$ $\endgroup$ – Cameron Buie Jul 15 '18 at 22:29
  • $\begingroup$ How about $g(n,s)\equiv1 , f(n,s)=\frac 1 {2^{n}}$? $\endgroup$ – Kavi Rama Murthy Jul 17 '18 at 0:25

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