Working Paper

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors


Grant Hillier, Raymond Kan, Xiaolu Wang

Published Date

11 June 2008


Working Paper (CWP14/08)

Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben, Hillier, Kan, and Wang). Typically, in a recursion of this type the k-th object of interest, dk say, is expressed in terms of all lower-order dj‘s. In Hillier, Kan, and Wang we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors.