Renormalization, Orthogonalization, and Generating Functions

Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , x, . . . } to get a sequence {Pn}∞n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of polynomials Pn, n ≥ 0. 1. Gaussian measure and Hermite polynomials Let μ be the Gaussian measure on the real line with mean 0 and variance σ, dμ(x) = 1 √ 2π σ e−x /2σ dx. (1.1) It is well-known that the Hermite polynomials