Zonal polynomial

In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.
They appear as zonal spherical functions of the Gelfand pairs
formula_1 (here, formula_2 is the hyperoctahedral group) and formula_3, which means that they describe canonical basis of the double class
algebras formula_4 and formula_5.
They are applied in multivariate statistics.
The zonal polynomials are the formula_6 case of the C normalization of the Jack function.