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I have been playing around with infinite series and I wondered if it is possible to find an expression for the series: $$ \sum_{k=0}^\infty x^{p(k)} $$ as a generalization of geometric series. $p(k)$ is an arbitrary polynomial.

Redundant Aunt
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  • As far as I know, (letting $x=10$ and $p=-k^2$), $1.1001000010000001\dots$ has no known closed form. It is transcendental (and irrational), I think. – Akiva Weinberger Oct 27 '14 at 16:21
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    It's impossible to find for arbitrary polynomial, but for some particular it's possible to find a closed forms. Probably in some elliptic theta functions. – m0nhawk Oct 27 '14 at 16:23
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    @m0nhawk I am interested in those particular cases. Do you know whether they are listed somewhere? (for example) – iago-lito Oct 19 '15 at 08:17
  • Do anyone know if such a series have a specific name ? – mikis Sep 14 '17 at 16:38

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