The formula is called the Bochner-Kodaira formula. It involves as all other Weitzenböck Formulars the curvature of the bundle, this time a holomorphic twisting bundle. Striking consequences are for example the Kodaira-Vanishing theorem.
You may take a look for example in Berline Getzler Vergne "Heat kernels and the dirac operator", page 135, Proposition 3.71 or in Lawson, Michelson "Spin geometry" Thm D.12
The proofs are always quite similar, you use that the symmetries of the curvature tensors and the clifford multiplication cancel each other to reduce the term which is not the connection laplace.