# JackPolynomials.jl documentation

Schur polynomials have applications in combinatorics and zonal polynomials have applications in multivariate statistics. They are particular cases of Jack polynomials. This package allows to evaluate these polynomials. It can also compute their symbolic form:

julia> jack = JackPolynomial(2, [3 ; 1], 2.5)
24.5x₁³x₂ + 14.0x₁²x₂² + 24.5x₁x₂³

julia> jack(1,2)
301.0

Such a polynomial is associated to a partition of an integer (the lambda argument in the Julia functions). It must be given as a decreasing vector of nonnegative integers (tail-zeros are allowed but they are ignored: the partion [3; 2; 0] is the same as [3; 2]).

## Reference

James Demmel & Plamen Koev. Accurate and efficient evaluation of Schur and Jack functions. Mathematics of Computation 75, 253 (2015), 223-239.

## Member functions

JackPolynomials.JackMethod
Jack(x, lambda, alpha)

Evaluates a Jack polynomial.

Arguments

• x: vector of real or complex numbers
• lambda: partition of an integer
• alpha: alpha parameter
source
JackPolynomials.JackPolynomialMethod
JackPolynomial(m, lambda, alpha)

Symbolic Jack polynomial. The coefficients of the polynomial will have the same type as alpha.

Arguments

• m: integer, the number of variables
• lambda: partition of an integer
• alpha: alpha parameter
source
JackPolynomials.SchurMethod
Schur(x, lambda)

Evaluates a Schur polynomial.

Arguments

• x: vector of real or complex numbers
• lambda: partition of an integer
source
JackPolynomials.SchurPolynomialMethod
SchurPolynomial(m, lambda[, type])

Symbolic Schur polynomial.

Arguments

• m: integer, the number of variables
• lambda: partition of an integer
• type: the type of the coefficients of the polynomial; default Int64
source
JackPolynomials.ZonalMethod
Zonal(x, lambda)

Evaluates a zonal polynomial.

Arguments

• x: vector of real or complex numbers
• lambda: partition of an integer
source
JackPolynomials.ZonalPolynomialMethod
ZonalPolynomial(m, lambda[, type])

Symbolic zonal polynomial.

Arguments

• m: integer, the number of variables
• lambda: partition of an integer
• type: the type of the coefficients of the polynomial; default Rational
source
JackPolynomials.ZonalQMethod
ZonalQ(x, lambda)

Evaluates a quaternionic zonal polynomial.

Arguments

• x: vector of real or complex numbers
• lambda: partition of an integer
source
JackPolynomials.ZonalQPolynomialMethod
ZonalQPolynomial(m, lambda[, type])

Symbolic quaternionic zonal polynomial.

Arguments

• m: integer, the number of variables
• lambda: partition of an integer
• type: the type of the coefficients of the polynomial; default Rational
source