In "The 1-2-3 of Modular Forms" Zagier gives, as an example of proposition 21 (pg 61), the identity

$$\vartheta_3(z)^2 = \sum_{n=0}^{\infty}\begin{pmatrix}2n\\n\end{pmatrix}\left(\frac{\lambda(z)}{16}\right)^n = F\left(\frac{1}{2},\frac{1}{2};1;\lambda(z)\right)$$

where $\lambda(z)$ is the Legendre modular form. Is there anywhere that gives a detailed derivation of how to get this? I have been trying to use the differential equation he gives in proof 1 $$ 0 = \sum_{n=0}^{k+1}(-1)^n\det\left(A_n(z)\right)D_t^nf(z)$$ Where $$A_n(z) = \begin{pmatrix} F_0 & \dots &\widehat{F_0D^n_tF_0} & \dots & F_0D^{k+1}_tF_0 \\ \vdots &&\vdots && \vdots \\ F_k & \dots & F_k D^n_tF_k & \dots & F_kD^{k+1}_tF_k \end{pmatrix}$$ and the $n^\text{th}$ column is removed and $F_n(z) = z^{k-n}f(z)$. However I do not see how to get this into a form which yields the proper differential equation. Are there any texts or papers that work this out explicitly?