I'm trying to find a closed form of $ S(n)= \sum_{i=0}^n a^{q^i} $ for positive constant params $a, q, n$. I searched it in google but I couldn't find any. I searched it in wolframalpha, but I didn't get any result.I thought there is some set like geometric progression but no results in google.

I tried finding more fundamental properties for the set $a_1,a_2...,a_n$ as $a_{k+1} = a_k^q$, but with not much of a success.

The one thing I managed to find is the sum $A(n) = \sum_{i=0}^n log(a^{q^i})$ which was trivial but still couldn't get any results with this sum in wolframalpha for some reason...

So my question is : Is it possible to find closed form of $S(n)$ and if so what method should I use to find it?


A theory of mine: If there is closed form function: $f(\sum_{i=0}^n x_i^q) = \sum_{i=0}^n x_i^{q^2} $ then $S(n+1) = f(S(n))+a$ however there might be set $Y$ where: $S(n) =\sum_{i=0}^n y_i^{q^n}$ then $S(n+1) = f(S(n)) + a$ is different from the old $S(n+1)$. So in order for $f(x)$ to exist $X$ must be a set of $a_1,a_2...,a_n$ elements and as such $f(x)$ can't be closed form and $S(n)$ respectively.

  • $\begingroup$ Welcome to Math SE! Take a look here for information about formatting your question with MathJax to make it more readable $\endgroup$ – DMcMor Sep 7 '17 at 14:18
  • $\begingroup$ @DMcMor I tried it but for some reason $\sum_{i=0}^n a^(q^i) $ didn't look good, so I thought it would be more clear using plain text. If you have any Ideas about the mathjax I should use please write it to me so I can fix my question's appearance $\endgroup$ – user2377766 Sep 7 '17 at 14:22
  • $\begingroup$ @user2377766 Try a^{q^i} instead it formats like this $a^{q^i}$ $\endgroup$ – kingW3 Sep 7 '17 at 14:24
  • $\begingroup$ @ kingW3 thanks $\endgroup$ – user2377766 Sep 7 '17 at 14:24
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    $\begingroup$ I guess you might add some conditions like $a>0$ etc $\endgroup$ – MAN-MADE Sep 7 '17 at 14:35

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