Jack function

Summary

Jack functions $J_\lambda^{(\alpha)}$ are symmetric functions indexed by an integer partition $\lambda$ and depending on a parameter $\alpha$. They are fixed by being triangular with respect to the monomial basis and orthogonal for a deformation of the Hall inner product. For $\alpha = 1$ they are multiples of the Schur polynomials, and for $\alpha = 2$ they are the zonal polynomials of multivariate statistics. They are the limit $q = t^\alpha$, $t \to 1$ of the Macdonald polynomials, and they are eigenfunctions of the Calogero–Sutherland model of quantum particles on a circle.