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SummaryFurther properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes the /c-class estimator in econometrics.

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The invariant polynomials (Davis [8] and Chikuse [2] with r(r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]).The ‘top order’ invariant polynomials , in which each of the partitions of ki 1 = 1,…,r, and has only one part, occur frequently in multivariate distribution theory (e.g., Hillier and Satchell [17] and Phillips [27]). In this paper we give three methods of constructing these polynomials, extending those of Ruben [28] for the top order zonal polynomials. The first two methods yield explicit formulae for the polynomials and then we give a recurrence procedure. It is shown that some of the expansions presented in Chikuse and Davis [4] are simplified for the top order invariant polynomials. A brief discussion is given on the ‘lowest order’ invariant polynomials.