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.
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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

3It'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

1@m0nhawk I am interested in those particular cases. Do you know whether they are listed somewhere? (for [example](http://math.stackexchange.com/questions/1483372/)) – iagolito Oct 19 '15 at 08:17

Do anyone know if such a series have a specific name ? – mikis Sep 14 '17 at 16:38