Bol identity for modular forms
= Bol identity for modular forms
{c}
{title2=$D^{1-k}:M_k^!\longrightarrow M_{2-k}^!$}
For integer $k<0$, the operator $D=(2\pi i)^{-1}d/dz=q\,d/dq$ 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.