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

<p><p>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 <i>k</i>-th object of interest, <i>d_k</i> say, is expressed in terms of all lower-order <i>d_j</i>'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.</p></p>