Demand for money

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 $(S_{2n},H_n)$ (here, $H_n$ is the hyperoctahedral group) and $(G{l}_n(\mathbb{ℝ}),O_n)$, which means that they describe canonical basis of the double class algebras $\mathbb{ℂ}[H_n\mathrm{\setminus }{S}_{2n}/H_n]$ and $\mathbb{ℂ}[O_d(\mathbb{ℝ})\mathrm{\setminus }{M}_d(\mathbb{ℝ})/O_d(\mathbb{ℝ})]$.

They are applied in multivariate statistics.

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

References

• Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.