  Let $n,m\ge 1$ and $\alpha>0$. We denote by $\mathcal{F}_{\alpha,m}$ the
$m$-analytic Bargmann--Segal--Fock space, i.e., the Hilbert space of all
$m$-analytic functions defined on $\mathbb{C}^n$ and square integrables with
respect to the Gaussian weight $\exp(-\alpha |z|^2)$. We study the von Neumann
algebra $\mathcal{A}$ of bounded linear operators acting in
$\mathcal{F}_{\alpha,m}$ and commuting with all ``horizontal'' Weyl
translations, i.e., Weyl unitary operators associated to the elements of
$\mathbb{R}^n$. The reproducing kernel of $\mathcal{F}_{1,m}$ was computed by
Youssfi [Polyanalytic reproducing kernels in $\mathbb{C}^n$, Complex Anal.
Synerg., 2021, 7, 28]. Multiplying the elements of $\mathcal{F}_{\alpha,m}$ by
an appropriate weight, we transform this space into another reproducing kernel
Hilbert space whose kernel $K$ is invariant under horizontal translations.
Using the well-known Fourier connection between Laguerre and Hermite functions,
we compute the Fourier transform of $K$ in the ``horizontal direction'' and
decompose it into the sum of $d$ products of Hermite functions, with
$d=\binom{n+m-1}{n}$. Finally, applying the scheme proposed by
Herrera-Ya\~{n}ez, Maximenko, Ramos-Vazquez [Translation-invariant operators in
reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we
show that $\mathcal{F}_{\alpha,m}$ is isometrically isomorphic to the space of
vector-functions $L^2(\mathbb{R}^n)^d$, and $\mathcal{A}$ is isometrically
isomorphic to the algebra of matrix-functions $L^\infty(\mathbb{R}^n)^{d\times
d}$.
