Derivative transformation of a weak modular form
= Derivative transformation of a weak modular form
{title2=$f^{(\ell)}(-1/z)$}
For a weight-$k$ <weak modular form> of level one,
$$
f^{(\ell)}(-1/z)=\sum_{j=0}^{\ell}\binom\ell j(k+j)_{\ell-j}z^{k+\ell+j}f^{(j)}(z),
$$
where $(a)_r=a(a+1)\cdots(a+r-1)$ and $(a)_0=1$. Differentiate and multiply by $z^2$ to obtain the coefficient recurrence $C_{\ell+1,j}=(k+\ell+j)C_{\ell,j}+C_{\ell,j-1}$.