How can I evaluate the following sum?

Question

While studying sequences and series, I came across summations of geometric series. I am able to derive the sum of a geometric progression and that of arithmetico–geometric sequence.

But taking a step further, I found the following series intriguing: the $i^{th}$ term is $a^{i^2}$ (instead of $a^{i}$ as in geometric series).

Is there a name for this series? How can we evaluate its sum?

$$ S = \sum_{i=0}^{n-1}a^{i^2} $$

First, I tried multiplying $S$ by $a$ and differentiating it by $a$, but to no avail. Then, I went through the eight PDF documents listed on Gould's home page containing enormous combinatorial identities, but I couldn't make any progress.

Answer 1

The corresponding infinite sum is related to the "incomplete theta function", $$\Theta_0(x,y)=\sum_{n\geq 0}x^ny^{n(n-1)/2}.$$ We have $$\sum_{n\geq 0}a^{n^2}=\Theta_0(a,a^2).$$ Accordingly, your series has a name "partial sum of the incomplete theta function".

Ref. A. Sokal, The leading root of the partial theta function, Adv. Math. 229 (2012), no. 5, 2603–2621,

and references there.

Such series were subject of a lot of research recently. On partial sums, see, for example

Katkova, Olga M.; Lobova, Tetyana; Vishnyakova, Anna M. On power series having sections with only real zeros. Comput. Methods Funct. Theory 3 (2003), no. 1-2, 425–441.