Is there any closed form known for the expression $\sum\_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!



Sum of series $a^{i^2}$
Calling your function $f(a),$ it is clear that $f(a)^{4} = \sum\_{n=1}^{\infty}r\_{4}(n) a^{n},$ where $r\_{4}(n)$ is the number of ways to express $n$ as a sum of four integer squares, as proved by Jacobi, who also gave an explicit description of $r\_{4}(n)$ in terms of the divisors of $n.$ I that sense ( and really regarding $f(a)$ as a formal power series in $a,$ which is a slight abuse), $f(a)$ is the fourth root of a "known" function.


