Let an entire function $F : \mathbb C \to \mathbb C$ of order $2$ be given by its canonical product $$ F(z)=z^me^{az^2+bz+c} \prod_{w \in Z} E_2(z/w), \quad n \in \mathbb N, $$ where $Z$ is the zero set of $F$ and where $E_2$ are the Weierstrass elementary factors of order $2$. Now let $p \in \mathbb C$. Then the shift of $F$ by $p$ is given by $$ z \mapsto F(z-p) = z^ke^{sz^2+tz+u} \prod_{w \in Z+p} E_2(z/w). $$ I'm wondering if there is a general formula for the constants $s,t$, and $u$ in the exponential factor of $F(z-p)$. In other words, how do the constants $s,t$, and $u$ arise from $Z,a,b$ and $c$?
Any help is highly appreciated!
