The dual multivariate Charlier and Edgeworth expansions

The Charlier differential series for distribution and density functions is the foundation for the Edgeworth expansions of distribution and density functions of sample estimators. Here, we give two forms of these expansions for multivariate distributions using multivariate Bell polynomials. Two forms arise because the multivariate Hermite polynomials have a dual form. These dual forms for the multivariate Charlier and Edgeworth expansions appear to be new.