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.
Definition The Jack function J κ ( α ) ( x 1 , x 2 , … , x m ) {\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})} of an integer partition κ {\displaystyle \kappa } , parameter α {\displaystyle \alpha } , and arguments x 1 , x 2 , … , x m {\displaystyle x_{1},x_{2},\ldots ,x_{m}} can be recursively defined as follows:
For m=1
J k ( α ) ( x 1 ) = x 1 k ( 1 + α ) ⋯ ( 1 + ( k − 1 ) α ) {\displaystyle J_{k}^{(\alpha )}(x_{1})=x_{1}^{k}(1+\alpha )\cdots (1+(k-1)\alpha )}
For m>1
J κ ( α ) ( x 1 , x 2 , … , x m ) = ∑ μ J μ ( α ) ( x 1 , x 2 , … , x m − 1 ) x m | κ / μ | β κ μ , {\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=\sum _{\mu }J_{\mu }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m-1})x_{m}^{|\kappa /\mu |}\beta _{\kappa \mu },}
where the summation is over all partitions μ {\displaystyle \mu } such that the skew partition κ / μ {\displaystyle \kappa /\mu } is a horizontal strip, namely
κ 1 ≥ μ 1 ≥ κ 2 ≥ μ 2 ≥ ⋯ ≥ κ n − 1 ≥ μ n − 1 ≥ κ n {\displaystyle \kappa _{1}\geq \mu _{1}\geq \kappa _{2}\geq \mu _{2}\geq \cdots \geq \kappa _{n-1}\geq \mu _{n-1}\geq \kappa _{n}} ( μ n {\displaystyle \mu _{n}} must be zero or otherwise J μ ( x 1 , … , x n − 1 ) = 0 {\displaystyle J_{\mu }(x_{1},\ldots ,x_{n-1})=0} ) and
β κ μ = ∏ ( i , j ) ∈ κ B κ μ κ ( i , j ) ∏ ( i , j ) ∈ μ B κ μ μ ( i , j ) , {\displaystyle \beta _{\kappa \mu }={\frac {\prod _{(i,j)\in \kappa }B_{\kappa \mu }^{\kappa }(i,j)}{\prod _{(i,j)\in \mu }B_{\kappa \mu }^{\mu }(i,j)}},}
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