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I found this question generalization of geometric series, which asks about generalization of geometric series. Unfortunately, there is no answer and seems there is no such things.

Anyhow, in my case I don't have a polynomial exponent, but a specific case with square root: $$\sum_{n=0}^\infty a^{\sqrt{n}},\ |a| < 1,$$

and in my case $a = e^{-t}$. Is there maybe some closed-form solution in terms of e.g. theta-function, or this is even more lost case than a polynomial one? Thanks.

pisoir
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  • I don't think there is any closed form for this sum, and the answers in the linked question agree. I would be excited to be proven wrong, though ^_^ – HallaSurvivor Dec 06 '20 at 09:15
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    @HallaSurvivor Thanks, I did not find that question. Yes, that will work. I close the question. – pisoir Dec 06 '20 at 09:28

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