Sum of increasing power

Question

What are the results of this: $$\sum_{i = 1}^{n}a^{i^{2}}$$ and this: $$\sum_{i = 1}^{n}a^{i^{3}}$$ expressed in terms of $n$, with $a$ being a predefined constant? Is there any general rule to calculating the sums of series like these for an arbitrary power of $i$?

Answer 1

For the first infinite series, what you want to learn about is Jacobi Theta Functions.

See, for example, the result in Wolfram|Alpha.

I don't know of anything corresponding to the second series.