na.numerical analysis - Is there any numerical technique to sum x^(n^alpha), n=0,1,...?

The $k$th power is the generating function for the ways of expressing $n$ as a sum of $k$ $alpha$th powers of positive integers. However, I don't think this helps much even for most integer values of $alpha$.

When $alpha = 1$, this is a geometric series.

When $alpha = 2$, this is a theta function related to the Jacobi triple product formula

$$prod_{m=1}^infty (1-x^{2m})(1+x^{2m-1}y^2)(1+x^{2m-1}y^{-2}) = sum_{n=-infty}^infty x^{n^2}y^{2n}$$
since
$$(frac 12 text{RHS}+frac12) bigg|_{y=1} = sum_{n=0}^infty x^{n^2} .$$

You may be able to compute the sum when $alpha=2$ more efficiently using one of the integral formulas or other properties for elliptic theta functions. Other than that, I don't know of special cases.

This doesn't say anything about whether some rapid series acceleration technique exists.

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