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^{-x^2+c_jx} 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 happens: 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 ∫ f(x) \exp(-x^2 + c_jx) 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.status: publishe