Simplify $\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$

Question

EDIT:

It is known that $$ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x} $$

Therefore, is it also possible to simplify $$\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$$ into something similar to the infinite geometric series sum?

Answer 1

In the case $x=1$, we have $$ \sum_{k=0}^\infty y^{k(k+1)/2} = \frac{\Theta_2(0, y^{1/2})}{2 y^{1/8}}$$ for $0 < |y| < 1$, where $\Theta_2$ is a Jacobi Theta function.

I do not know of a "closed-form" expression for general $x$, but it's certainly not going to be an elementary function.