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 ${\displaystyle (S_{2n},H_{n})}$ (here, ${\displaystyle H_{n))$ is the hyperoctahedral group) and ${\displaystyle (Gl_{n}(\mathbb {R} ),O_{n})}$, which means that they describe canonical basis of the double class algebras ${\displaystyle \mathbb {C} [H_{n}\backslash S_{2n}/H_{n}]}$ and ${\displaystyle \mathbb {C} [O_{d}(\mathbb {R} )\backslash M_{d}(\mathbb {R} )/O_{d}(\mathbb {R} )]}$.

They are applied in multivariate statistics.

The zonal polynomials are the ${\displaystyle \alpha =2}$ case of the C normalization of the Jack function.