I.—A class of symmetric polynomials with a parameter

Abstract

In an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).