  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.
