A General Expression for Hermite Expansions with Applications.

Hermite polynomials arise when dealing with functions of normally distributed variables, and are commonly thought of as the analog of the simple polynomials on functions of regular variables. Therefore the Hermite expansion should be an analog of the Taylor expansion. Indeed there is a strong connection between the two - the general coefficient in the Hermite expansion is the weighted integral of the nth derivative, as compared to the nth derivative evaluated at zero in the case of Taylor. This fact can be used to derive the Hermite expansion for the integral and the derivative of a function. Furthermore, it provides a method of providing simple proofs of many of the Hermite identities. This connection is used to derive the Hermite expansions of the normal probability distribution function, the normal cumulative distribution function and the indicator function. Finally, an algorithm to numerically perform a Hermite expansion is presented, which is efficient in the sense that is only requires a single call to a quadrature method. [ABSTRACT FROM AUTHOR]