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Multivariate Orthogonal Polynomials

In the symmetric framework of Dumitriu et al. (2007), multivariate orthogonal polynomials p_kappa^((alpha))(x_1,...,x_m) are indexed by partitions kappa and orthonormal with respect to an m-dimensional inner product of the form

int_(I^m)p_kappa^((alpha))(x_1,...,x_m)p_mu^((alpha))(x_1,...,x_m)product_(1<=i<j<=m)|x_i-x_j|^(2/alpha)product_(i=1)^mw(x_i)dx_1...dx_m=delta_(kappamu).

Here I is an interval, w is a weight, alpha>0, and delta is the Kronecker delta. The factor involving pairwise differences couples the variables and disappears when m=1.

Taking I=R and w(x)=e^(-x^2/2) gives the multivariate Hermite polynomials. Taking I=[0,infty) and w(x)=x^gammae^(-x) gives the multivariate Laguerre polynomials. Taking I=[0,1] and w(x)=x^(g_1)(1-x)^(g_2) gives the multivariate Jacobi polynomials. These families may be expanded in Jack polynomials and have applications to random matrix eigenvalue statistics.

See also

Jack Polynomial, Multivariate Hermite Polynomial, Multivariate Hypergeometric Function, Multivariate Jacobi Polynomial, Multivariate Laguerre Polynomial, 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 Orthogonal Polynomials." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultivariateOrthogonalPolynomials.html

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