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We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P ( a Y b ) for Y ~ N p (0, V ) and for repeated integrals of the density of  Y . When V −1 y > 0 in R 3 the expansion for P ( Y y ) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of N p (0, V −1 ). This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials. These are given explicitly.

Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal