Three Specific Orthogonal Series Representations

In this section, the parameters of equation (1) are specified that result in the Gram-Charlier, Gamma, and Beta series. Let

where is the expectation operator, in this case, with respect to the density .

Table 1. PDF and CDF series parameters for the Gram-Charlier, Gamma, and Beta series.

Table 1 lists the probability density and distribution series parameters for the specific cases of interest here. Note that , , and are the Hermite, Generalized Laguerre, and Jacobi orthogonal polynomials of degree as defined in [2] with respect to the corresponding weights, , above. (The Hermite and Jacobi polynomials used here are defined with respect to different weight functions than the built-in Mathematica functions for these polynomial types.) The weight functions used here assure that the first term of the Gram-Charlier, Gamma, or Beta series is the Gaussian, Gamma, or Beta density function, respectively, as shown in the final row of the table. This first term is referred to as the base probability density function in the remainder.