The invariant polynomials (Davis  and Chikuse  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  and Phillips ).
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  and Phillips ). In this paper we give three methods of constructing these polynomials, extending those of Ruben  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  are simplified for the top order invariant polynomials. A brief discussion is given on the ‘lowest order’ invariant polynomials.