Let C κ(S) be the zonal polynomial of the symmetric m×m matrix S=(sij), corresponding to the partition κ of the non-negative integer k. If ∂/∂S is the m×m matrix of differential operators with (i, j)th entry ((1+δij)∂/∂sij)/2, δ being Kronecker's delta, we show that Ck(∂/∂S)Cλ(S)=k!δλkCk(I), where λ is a partition of k. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.
|Original language||English (US)|
|Number of pages||3|
|Journal||Annals of the Institute of Statistical Mathematics|
|State||Published - Dec 1982|
All Science Journal Classification (ASJC) codes
- Statistics and Probability