2\left(z\frac{\partial}{\partial z}-{\overline{z}}\frac{\partial}{\partial{\overline{z}}}\right)-{\lvertz\rvert}^2\right\}$$ is the Hamiltonian of a nonrelativistic quantum particle moving on the plane under the action of a constant external magnetic field applied perpendicularly to the plane. The functions $e^{-{\lvertz\rvert}^2/2}h_{m,n}(z,{\overline{z}})$ are the eigenfunctions of $\mathfrak{L}$ with eigenvalues $n+\frac{1}{2}$, each eigenvalue being infinitely degenerate. The degeneracy is given by $m=0,1,2,\cdots$. For details see [@Ghan; @Mat].\

Quaternionic Hermite polynomials {#sec-quherm-polyn}
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In this section we define the quaternionic Hermite polynomials $H_n({\mathbf{q}})$ and $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$, by analogy with the complex polynomials introduced above. We shall identify the set of all polynomials $H_n({\mathbf{q}})$ as an orthogonal set in a $L^2$-space and similarly for the $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$. We also obtain reproducing kernels using the polynomials $H_n({\mathbf{q}})$ and $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ and the corresponding reproducing kernel Hilbert spaces. Finally we look at an operator $\mathfrak{L}_H$ which can be considered as the quaternionic version of the Landau operator (\[E6\]).

It is well-known (see, e.g., [@Thi]) that any ${\mathbf{q}}\in H$, in the $2\times 2$ matrix representation, can be written as $$\label{dc}
{\mathbf{q}}=u_{{\mathbf{q}}}Zu_{{\mathbf{q}}}^{\dagger},$$ where $$u_{{\mathbf{q}}}=\left(\begin{array}{cc}
e^{i\psi/2}&0\\
0&e^{-i\psi/2}\end{array}\right)
\left(\begin{array}{cc}
\cos\frac{\phi}{2}&i\sin\frac{\phi}{2}\\
i\sin\frac{\phi}{2}&\cos\frac{\phi}{2}\end{array}\right)
\left(\begin{array}{cc}
e^{i\psi/2}&0\\
0&e^{-i\psi/2}\end{array}\right)\in SU(2),$$ and $Z=\left(\begin{array}{cc}
z&0\\
0&{\overline{z}}\end{array}\right), \; z \in \mathbb C$. Since $SU(2)$ is a compact group, let $d\omega(u_{{\mathbf{q}}})$ be the normalized Haar measure on it. From the decomposition (\[dc\]), 