My understanding is that the Jacobi Theta function is fundamental solution of heat equation:
$\displaystyle \vartheta (x,it)=1+2\sum _{n=1}^{\infty }\exp \left(-\pi n^{2}t\right)\cos(2\pi nx)$
The following heat kernel is also fundamental solution of heat equation:
$\Phi (x,t)={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)$
But I do not see how to expand above heat kernel to get Jacobi Theta function.
Can anyone provide some hints on how to expand above heat kernel to get Jacobi Theta function ?
Thank you.