I have asked this question two days ago. One helpful person in the comments directed me to the Jacobi Theta functions and I was able to solve the rest myself. Starting from that point, I have:
$\sum_{i=0}^{\infty}q^{(2i+1)^2}=1+[\vartheta(q;0)-\vartheta(q^4;0)]/2$,
where $q=\exp(-a)$ Then,
$q\frac{\partial}{\partial{q}}\sum_{i=0}^{\infty}q^{(2i+1)^2}=\sum_{i=0}^{\infty}q^{(2i+1)^2}(2i+1)^2$.
So I only need to take the partial derivative of the first equation to find the result. Doing so gives me
$q\frac{\partial}{\partial{q}}\sum_{i=0}^{\infty}q^{(2i+1)^2}=\frac{q}{2}[\frac{\partial\vartheta(q;0)}{\partial q}-\frac{\partial}{\partial q}\vartheta(q^4;0)]$
$q\frac{\partial}{\partial{q}}\sum_{i=0}^{\infty}q^{(2i+1)^2}=\frac{q}{2}[\frac{\partial\vartheta(q;0)}{\partial q}-\frac{\partial\vartheta(q^4;0)}{\partial q}(4q^3)]$
Finally, using the relation described here, for python implementation, the last result is
$q\frac{\partial}{\partial{q}}\sum_{i=0}^{\infty}q^{(2i+1)^2}=\frac{q}{2}[-\frac{\partial^2\vartheta(q;0)}{(4q)\partial z^2}+\frac{\partial^2\vartheta(q^4;0)}{(4q)\partial z^2}(4q^3)]$.
However, while I verified the initial equation in python, I can't verify this final result.
Where am I going wrong?
Thanks!