Derivative transformation of a weak modular form ($f^{(\ell )}(-1/z)$)

For a weight-$k$ weak modular form of level one,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 $C_{\ell +1,j}=(k+\ell +j)C_{\ell ,j}+C_{\ell ,j-1}$.

For integer $k<0$, the operator $D=(2\pi i{)}^{-1}d/dz=qd/dq$ sends a weight-$k$ level-one weak modular form to a weight-$2-k$ one after $1-k$ iterations. In the derivative transformation of a weak modular form, every lower derivative coefficient then contains a zero factor. This also holds on the subspace $M_k^{!}$ of forms meromorphic at the cusp.