The multivariate Hermite polynomials 
are symmetric
polynomials indexed by a partition 
and orthogonal on 
with respect to the weight
![W_H(x_1,...,x_m)=exp[-1/2sum_(i=1)^mx_i^2]product_(1<=i<j<=m)|x_i-x_j|^(2/alpha),](/images/equations/MultivariateHermitePolynomial/NumberedEquation1.svg) |
where 
(Dumitriu et al. 2007). When 
, the product over pairs disappears and the family reduces,
up to normalization, to the Hermite polynomials
associated with the Gaussian weight 
. Multivariate Hermite polynomials also occur in eigenvalue statistics for Hermite random
matrix ensembles, so called because their joint eigenvalue density is proportional
to the Hermite weight 
.
