  Motivated by the concept of "generating operators" for a countable family of
operators introduced in the recent paper (arXiv:2306.16800), we find a method
to reconstruct the Rankin--Cohen brackets from a very simple multivariable
contour integral, and obtain a new proof of their covariance. We also establish
a closed formula of the "generating operator" for the Rankin--Cohen brackets in
full generality.
