Generating Functions and Short Recursions, with Applications to the Moments of Quadratic Forms in Noncentral Normal Vectors

Grant Hillier, Raymond Kan, Xiaolu Wang
2009 Social Science Research Network  
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, d k say, is expressed in terms of all lower-order d j 's. In Hillier, Kan, and Wang [9] we pointed out that, in the case of top-order zonal polynomials (and generalizations
more » ... 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.
doi:10.2139/ssrn.1410755 fatcat:klebor5xofbi5jbamkfmeh464e