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Multivariate Jacobi Polynomial

The multivariate Jacobi polynomials J_kappa^((alpha,g_1,g_2))(x_1,...,x_m) are obtained by applying Gram-Schmidt orthonormalization to the Jack polynomials on [0,1]^m with respect to the weight

W_J(x_1,...,x_m)=product_(i=1)^mx_i^(g_1)(1-x_i)^(g_2)product_(1<=i<j<=m)|x_i-x_j|^(2/alpha),

where alpha>0 and g_1,g_2>-1 (Dumitriu et al. 2007). When m=1, the product over pairs disappears and an affine transformation of the variable gives the ordinary Jacobi polynomials.

See also

Jack Polynomial, Jacobi Polynomial, Multivariate Laguerre Polynomial, Multivariate Orthogonal Polynomials

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References

Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." J. Symb. Comput. 42, 587-620, 2007. https://doi.org/10.1016/j.jsc.2007.01.005.

Cite this as:

Weisstein, Eric W. "Multivariate Jacobi Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultivariateJacobiPolynomial.html

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