Multiple Hermite polynomials and simultaneous Gaussian quadrature

Abstract

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 $w_j(x)=e^{-x^2+c_{j}x}$ with different means $c_j/2$, $1\le j\le r$. 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 $c_j$, $1\le j\le r$, the zeros may accumulate on $s$ disjoint intervals, where $1\le s\le r$. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form ${\int }_{-\infty }^{\infty }f(x)\exp (-x^2+c_{j}x)dx$ simultaneously for $1\le j\le r$ 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.

18 pages, 5 figures, 1 table

References in corpus (1)

• Simultaneous Gaussian quadrature for Angelesco systems

Cited by in corpus (3)

• Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
• A Golub-Welsch version for simultaneous Gaussian quadrature
• Christoffel functions for multiple orthogonal polynomials