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

Abstract. 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 ∼ Np(0, V ) and for repeated integrals of the density of Y . When V −1y > 0 in R 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 Np(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.