I am trying to find the closed form of generating function such that $x^1+x^4 + x^9 +x^{16} +x^{25} + x^{36} +x^{49} +x^{64}+....$
As you realize , their exponentials are the sequence of $1,4,9,16,..., n^2,...$ where $n$ is positive integer. I do not know what i should do , i tried to put some values in $\frac{1}{1-x}$ but it does not satisfy.
NOTE= I am working over generating functions , a solution was suggested but theta function does not satisfy me . I need a closed form like $\frac{x^3}{1-x^3}$ etc. (It is given randomly , $\frac{x^3}{1-x^3}$ does not related to the question )
This is series representation of the famous Jacobi theta function. And in your case this corresponds to the subscript $3$ and hence this sum becomes as follows : $$\sum \limits_{k=1}^\infty x^{k^2}=\frac{1}{2}\left( \vartheta_3 (0,x) -1\right) $$ for $|x|<1$ Note : before actually knowing this much about the function I kind of played with series a lot and was curious to derive ample of identities (approximations actually which I realized afterwards ) similar to this example given : $$\sum \limits_{k=1}^{\infty } \frac{1}{e^{k^2}}\approx \frac{\sqrt{\pi } -1}{2}$$ But noticing the first few decimal digits I almost was moved by the fact that this result is correct which isn't after I came to know about this function.