On the expansion of \(C^*_{\rho}(V+I)\) as a sum of zonal polynomials
The coefficients \(a_{\tau \rho}\), sometimes called ''generalized binomial coefficients'' in the expansion \(C^*_{\rho}(V+I)=\sum_{\tau}a_{\rho \tau}C^*_{\tau}(V)\), are computed explicitly when \(t=r+1\), where \(\rho\) is a partition of r and \(\tau\) a partition of t. A recursion formula permits the calculation of the general \(a_{\tau \rho}\). Several properties of \(a_{\tau \rho}\) are proved. A connection between the \(a_{\tau \rho}\) 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.
scientific article; zbMATH DE number 3936262
scientific article