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 (here, is the hyperoctahedral group) and 
, which means that they describe canonical basis of the double class algebras and ![\mathbb{C}[O_d(\mathbb{R})\backslash
M_d(\mathbb{R})/O_d(\mathbb{R})]](http://upload.wikimedia.org/math/e/8/d/e8d986faf136f16b2f3f9e1b434acfa6.png)
.
They are applied in multivariate statistics.
The zonal polynomials are the case of the C normalization of the Jack function.