Closed form for the summation $\sum_{k=1}^n\frac{1}{r^{k^2}}$

Question

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this type of finite series. But I have no idea about attempt to this problem. Thank you.

Answer 1

We have

$$\sum_{k=1}^2 1/r^{k^2}= (r^3+1)/r^4$$ $$\sum_{k=1}^3 1/r^{k^2}= (r^8+r^5+1)/r^9$$ $$\sum_{k=1}^4 1/r^{k^2}= (r^{15}+r^{12}+r^7+1)/r^{16}$$ $$\sum_{k=1}^5 1/r^{k^2}= (r^{24}+r^{21}+r^{16}+r^9+1)/r^{25}$$

So, for a fixed $n$ we have

$$\sum_{k=1}^n 1/r^{k^2}= w_{n-1}(r)/r^{n^2}$$

where $w_{n-1}(r)$ is a polynomial of degree $n^2-1$ in $r$. But obtaining the formula when $n$ goes to infinity is impossible as hinted in the above comments.