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

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 [19], Hillier, Kan, and Wang [9]). 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 [9] 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 [9] 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.