In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a zonal polynomial is a multivariate symmetric

Symmetric polynomial

In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...

 homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

. The zonal polynomials form a basis of the space of symmetric polynomials.

They appear as zonal spherical function

Zonal spherical function

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G...

s of the Gelfand pair

Gelfand pair

In mathematics, the expression Gelfand pair is a pair consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to...

s
(here, is the hyperoctahedral group) and , which means that they describe canonical basis of the double class
algebras and .

They are applied in multivariate statistics

Multivariate statistics

Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...

.

The zonal polynomials are the case of the C normalization of the Jack function

Jack function

In mathematics, the Jack function, introduced by Henry Jack, is a homogenous, symmetric polynomial which generalizes the Schur and zonal polynomials,and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.-Definition:...

.