Expressions for the normal distribution and repeated normal integrals

We give a new expression for Mills ratio and five new expressions for repeated integrals of the univariate normal density, or equivalently for the Hermite functions. The Hermite functions are shown to be the negative moments of x+iNx+iN where N   is a unit normal random variable and i=-1. This extends an earlier result that the Hermite polynomials are the positive moments of x+iNx+iN. We also give the derivatives of Mills’ ratio and its inverse.