# Multiple Hermite polynomials and simultaneous Gaussian quadrature

## TL;DR

This paper studies multiple Hermite polynomials, their properties, zero distributions, and uses their zeros for simultaneous Gaussian quadrature to approximate multiple integrals with Gaussian weights.

## Contribution

It introduces properties and asymptotic zero distributions of multiple Hermite polynomials and applies their zeros to develop a new simultaneous quadrature method for Gaussian-weighted integrals.

## Key findings

- Zeros may accumulate on multiple disjoint intervals depending on parameters.
- Asymptotic zero distribution is characterized for different parameter regimes.
- A new simultaneous Gaussian quadrature method is proposed for multiple integrals.

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