On the expansion of Cϱ∗(V + I) as a sum of zonal polynomials

AbstractThe coefficients aτϱ, sometimes called “generalized binomial coefficients” in the expansion Cϱ∗(V +I) = ΣτaϱτCτ∗(V), are computed explicitly when t = r + 1, where ϱ is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general aτϱ. Several properties of aτϱ are proved. A connection between the aτϱ and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials