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 formula_1 
of an integer partition formula_2, parameter formula_3, and arguments formula_4 can be recursively defined as 
follows:
where the summation is over all partitions formula_7 such that the skew partition formula_8 is a horizontal strip, namely
where formula_13 equals formula_14 if formula_15 and formula_16 otherwise. The expressions formula_17 and formula_18 refer to the conjugate partitions of formula_2 and formula_7, respectively. The notation formula_21 means that the product is taken over all coordinates formula_22 of boxes in the Young diagram of the partition formula_2.
Combinatorial formula.
In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials formula_24 in "n" variables:
The sum is taken over all "admissible" tableaux of shape formula_26 and 
with 
An "admissible" tableau of shape formula_29 is a filling of the Young diagram formula_29 with numbers 1,2,…,"n" such that for any box ("i","j") in the tableau,
This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials.
C normalization.
The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product:
This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as
where
For formula_41 is often denoted by formula_42 and called the Zonal polynomial.
P normalization.
The "P" normalization is given by the identity formula_43, where 
where formula_45 and formula_46 denotes the arm and leg length respectively. Therefore, for formula_47 is the usual Schur function.
Similar to Schur polynomials, formula_48 can be expressed as a sum over Young tableaux. However, one need to add an extra weight to each tableau that depends on the parameter formula_3.
Thus, a formula for the Jack function formula_50 is given by
where the sum is taken over all tableaux of shape formula_29, and formula_53 denotes the entry in box "s" of "T".
The weight formula_54 can be defined in the following fashion: Each tableau "T" of shape formula_29 can be interpreted as a sequence of partitions 
where formula_57 defines the skew shape with content "i" in "T". Then 
where 
and the product is taken only over all boxes "s" in formula_29 such that "s" has a box from formula_61 in the same row, but "not" in the same column.
Connection with the Schur polynomial.
When formula_62 the Jack function is a scalar multiple of the Schur polynomial
where
is the product of all hook lengths of formula_2.
Properties.
If the partition has more parts than the number of variables, then the Jack function is 0:
Matrix argument.
In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If formula_67 is a matrix with eigenvalues
formula_4, then 