Jack function
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 of an integer partition , parameter , and arguments can be recursively defined as follows: For m=1 For m>1 where the summation is over all partitions such that the skew partition is a horizontal strip, namely ( must be zero or otherwise ) and where equals if and otherwise. The expressions and refer to the conjugate partitions of and , respectively. The notation means that the product is taken over all coordinates of boxes in the Young diagram of the partition . Combinatorial formula In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials in n variables: The sum is taken over all admissible tableaux of shape and with An admissible tableau of shape is a filling of the Young diagram with numbers 1,2,…,n such that for any box (i,j) in the tableau, whenever whenever and A box is critical for the tableau T if and This