MULTIPLE HERMITE POLYNOMIALS AND SIMULTANEOUS GAUSSIAN QUADRATURE

Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to r > 1 normal (Gaussian) weights w(j)(x) = e(-x2+cjx) with different means c(j)/2, 1 <= j <= r. These polynomials have a number of properties such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated, and an interesting new feature is observed: depending on the distance between the c(j), 1 <= j <= r, the zeros may accumulate on s disjoint intervals, where 1 <= s <= r. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form integral(infinity)(-infinity)f (x) exp( -x(2) + c(j)x) dx simultaneously for 1 <= j <= r for the case r = 3 and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights.