Jack function について

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Jack polynomial ：ウィキペディア英語版

Jack function
In mathematics, the Jack function, introduced by Henry Jack, 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_\kappa^(x_1,x_2,\ldots,x_m)

\kappa

, parameter

\alpha

and
arguments

x_1,x_2,\ldots,

can be recursively defined as
follows:
; For ''m''=1 :
:

J_^(x_1)=x_1^k(1+\alpha)\cdots (1+(k-1)\alpha)

; For ''m''>1:
:

J_\kappa^(x_1,x_2,\ldots,x_m)=\sum_\muJ_\mu^(x_1,x_2,\ldots,x_)x_m^\beta_,

where the summation is over all partitions

\mu

such that the skew partition

\kappa/\mu

is a horizontal strip, namely
:

\kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_\ge\mu_\ge\kappa_n

(

\mu_n

must be zero or otherwise

J_\mu(x_1,\ldots,x_)=0

) and
:

\beta_=\frac^\kappa(i,j)}^\mu(i,j)},

where

B_^\nu(i,j)

equals

\kappa_j'-i+\alpha(\kappa_i-j+1)

\kappa_j'=\mu_j'

and

\kappa_j'-i+1+\alpha(\kappa_i-j)

otherwise. The expressions

\kappa'

and

\mu'

refer to the conjugate partitions of

\kappa

and

\mu

, respectively. The notation

(i,j)\in\kappa

means that the product is taken over all coordinates

(i,j)

of boxes in the Young diagram of the partition

\kappa

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