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I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.

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For $l\geq 3$, I believe the sum cannot be expressed in terms of standard special functions.

For $l=2$, the answer is (almost by definition) $$\frac{1+\vartheta_3(a)}{2},$$ where $\vartheta_3(a)=\vartheta_3(0,a)$ denotes the Jacobi theta function of zero argument (theta constant).

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