I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for $|\omega_1|\rightarrow 0$, to find an expansion of the form $$ \wp(z,\omega_1,\omega_2) = \sum_{n=-N}^{\infty} F_n(z,\omega_1) B_n(\omega_1) $$ and similar for $\zeta$, where I would be happy with the basis of functions being either $B_n( \omega_1) = \omega_1^n$ or $B_n(\omega_1) = \exp(-n/\omega_1)$ or possibly a combination of the two. Here, $N$ is an integer and the $F_n$ are the coefficients.
I have tried finding this online and tried to use several trigonometric expansions to get this form, but failed until now. In particular, I used the Digital library of mathematical functions:
Is there anything known about expansions of this type?