Is there a "hyper-geometric" or "super-geometric" sum formula?

Asked Oct 11 '23 at 14:43

Active Oct 25 '23 at 15:18

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I know that when dealing with sums of the form $S_n=\sum_{k=1}^n q^k$ with $q\neq 1$ we have the formula $S_n= \frac{r^n-1}{r-1}$. But is there a known closed formula for a sum $S_n':=\sum_{k=1}^n q^{q^k}$?

Does anyone know of a solution or a useful relevant concept? I have a feeling that I am simply overlooking a simple solution here.

Later edit

Following the comments, I realize that there is no known closed formula for the sum. I was wondering however whether there are effective estimates on the sum. Specifically, how does the tail part of the sum behaves. If we take $n\ll N$, are there good estimates on $\frac{S_{n}'}{S_N'}$?

I Think that this term should converge to zero very quickly.

edited Oct 25 '23 at 15:18

asked Oct 11 '23 at 14:43

Keen-ameteur

7,663

2

I am pretty sure that there is no known closed form. Even in the case , say , $q=2$. – Peter Oct 11 '23 at 14:51
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Neither Wolfram Alpha nor Mathematica come up with a closed formula. – irchans Oct 11 '23 at 15:20
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Note that "hypergeometric" already has a couple standard meanings, e.g. series/functions and distributions. – Noah Schweber Oct 11 '23 at 16:03
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See Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$. – Dave L. Renfro Oct 11 '23 at 17:25

Is there a "hyper-geometric" or "super-geometric" sum formula?

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