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Discussion Paper 0918, "Generating Functions and Short Recursions, with Applications to the Moments of Quadratic Forms in Noncentral Normal Vectors", by Grant Hillier, Raymond Kan & Xiaoulu Wang

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 (1962), Hillier, Kan, and Wang (2009)). 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 (2009) 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 Hillier, Kan, and Wang (2009) 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.

Keywords: Generating Functions, Invariant Polynomials, Non-central Normal Distribution, Recursions, Symmetric Functions, Zonal Polynomials.

JEL Classification: C01, C46, C63