$\alpha,\beta >0$ $f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$
$\frac{\partial{f(x,z)}}{\partial z}=\beta \sum \limits_{n=1}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{(n-1)!}$
$\frac{\partial{f(x,z)}}{\partial z}|_{z=2 \frac{ \alpha}{\beta} x+ z_1}=\beta \sum \limits_{n=1}^\infty \frac{e^{-\alpha n^2 x+\beta n (2 \frac{ \alpha}{\beta} x+ z_1)}}{(n-1)!}$
$\frac{\partial{f(x,z)}}{\partial z}|_{z=2 \frac{ \alpha}{\beta} x+ z_1}=\beta e^{\alpha x+ \beta z_1} \sum \limits_{n=1}^\infty \frac{e^{-\alpha (n-1)^2 x+\beta (n-1) z_1}}{(n-1)!}$
$\frac{\partial{f(x,z)}}{\partial z}|_{z=2 \frac{ \alpha}{\beta} x+ z_1}=\beta e^{\alpha x+ \beta z_1} \sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z_1}}{n!}$
$\frac{\partial{f(x,z)}}{\partial z}|_{z=2 \frac{ \alpha}{\beta} x+ z_1}=\beta e^{\alpha x+ \beta z_1} f(x,z_1)$
I do not know how to solve this kind differential equations.
Do you know how to solve that?
Can we express the function as known functions such as Jacobi Theta Functions etc?
Also could you please share your knowledge about the function if you know it.
Thanks a lot for answers
EDIT:
Another property is:
$-\alpha\frac{\partial^2{f(x,z)}}{\partial z^2}=\beta^2 \frac{\partial{f(x,z)}}{\partial x} $