In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.

The Jack function ${\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})}$ of an integer partition ${\displaystyle \kappa }$, parameter ${\displaystyle \alpha }$, and arguments ${\displaystyle x_{1},x_{2},\ldots ,x_{m}}$ can be recursively defined as follows:

where the summation is over all partitions ${\displaystyle \mu }$ such that the skew partition ${\displaystyle \kappa /\mu }$ is a horizontal strip, namely

where ${\displaystyle B_{\kappa \mu }^{\nu }(i,j)}$ equals ${\displaystyle \kappa _{j}'-i+\alpha (\kappa _{i}-j+1)}$ if ${\displaystyle \kappa _{j}'=\mu _{j}'}$ and ${\displaystyle \kappa _{j}'-i+1+\alpha (\kappa _{i}-j)}$ otherwise. The expressions ${\displaystyle \kappa '}$ and ${\displaystyle \mu '}$ refer to the conjugate partitions of ${\displaystyle \kappa }$ and ${\displaystyle \mu }$, respectively. The notation ${\displaystyle (i,j)\in \kappa }$ means that the product is taken over all coordinates ${\displaystyle (i,j)}$ of boxes in the Young diagram of the partition ${\displaystyle \kappa }$.

In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials ${\displaystyle J_{\mu }^{(\alpha )}}$ in n variables:

The sum is taken over all admissible tableaux of shape ${\displaystyle \lambda ,}$ and

with

An admissible tableau of shape ${\displaystyle \lambda }$ is a filling of the Young diagram ${\displaystyle \lambda }$ with numbers 1,2,…,n such that for any box (i,j) in the tableau,