$d\mu ({\mathbf{q}}) := e^{-|z|^2}d\lambda d\omega(u_{{\mathbf{q}}})$ is a measure on $H$ and $L^2_H(H, e^{-|z|^2}d\lambda d\omega(u_{{\mathbf{q}}}))$ can be considered to be a [*left quaternionic Hilbert space*]{}, with an inner product defined as in (\[left-sc-prod\]).

From (\[dc\]), for any $i,j\in\mathbb{N}$ we have $${\mathbf{q}}^i=u_{{\mathbf{q}}}\left(\begin{array}{cc}
z^i&0\\
0&{\overline{z}}^i\end{array}\right)u_{{\mathbf{q}}}^{\dagger}\quad\text{and}\quad
{\overline{{\mathbf{q}}}}^j=u_{{\mathbf{q}}}\left(\begin{array}{cc}
{\overline{z}}^j&0\\
0&z^j\end{array}\right)u_{{\mathbf{q}}}^{\dagger}$$ and thereby, $$\label{E7}
{\mathbf{q}}^i\overline{{\mathbf{q}}}^j=u_{{\mathbf{q}}}\left(\begin{array}{cc}
z^i{\overline{z}}^j&0\\
0&{\overline{z}}^iz^j\end{array}\right)u_{{\mathbf{q}}}^{\dagger}.$$

The quaternionic Hermite polynomials $H_n({\mathbf{q}})$
--------------------------------------------------------

Let $d\eta({\mathbf{q}})=d\mu_s(z)d\omega(u_{{\mathbf{q}}})$, with $d\mu_s$ as in (\[s-measure\]). Since $${\mathbf{q}}^{n-2m}=u_{{\mathbf{q}}}\left(\begin{array}{cc}z^{n-2m}&0\\0&\overline{z}^{n-2m}\end{array}\right)
u_{{\mathbf{q}}}^{\dagger}$$ and the $C_{nm}$ in (\[H1\]) are real numbers, we have $$C_{nm}{\mathbf{q}}^{n-2m}=u_{{\mathbf{q}}}\left(\begin{array}{cc}C_{nm}z^{n-2m}&0\\0&C_{nm}\overline{z}^{(n-2m)}
\end{array}\right)u_{{\mathbf{q}}}^{\dagger}$$ and thereby $$n!\sum_{m=0}^{[n/2]}C_{nm}{\mathbf{q}}^{n-2m}=u_{{\mathbf{q}}}\left(\begin{array}{cc}n!\sum_{m=0}^{[n/2]}
C_{nm}z^{n-2m}&0\\0&n!\sum_{m=0}^{[n/2]}C_{nm}\overline{z}^{(n-2m)}\end{array}\right)
u_{{\mathbf{q}}}^{\dagger}$$ That is $$H_n({\mathbf{q}})=u_{{\mathbf{q}}}\left(\begin{array}{cc}H_n(z)&0\\0&H_n(\overline{z})\end{array}\right)
u_{{\mathbf{q}}}^{\dagger}.$$ Observe that $$\overline{H_n({\mathbf{q}})}=H_n(\overline{{\mathbf{q}}})=u_{{\mathbf{q}}}\left(\begin{array}{cc}H_n(\overline{z})&0\\
0&H_n(z)\end{array}\right)u_{{\mathbf{q}}}^{\dagger}.$$

Similarly, from (\[H1\]) we easily see that $$H({\mathbf{q}}) = n!\sum_{m=0}^{[n/2]}\frac{(-1)^m(2{\mathbf{q}})^{n-2m}}{m!(n-2m)!}\;,
\label{q-herm-exp}$$ which means that they satisfy the same recursion relations, $${\mathbf{q}}H_n ({\mathbf{q}}) = \frac 12 H_{n+1}({\mathbf{q}}) + n H_{n-1} ({\mathbf{q}})\; ,
\label{q-herm-rec}$$ as the real Hermite polynomials $H_n (x)$, or also $$H_{n+1} ({\mathbf{q}}) = 2{\mathbf{q}}H_n({\mathbf{q}}) - H^\prime_n({\mathbf{q}})\; ,
\label{q-herm-rec2}$$ the prime denoting the (Cullen) derivative.

Let $$L_H^2(H,d\eta({\mathbf{q}}))=\left\{f:H\rightarrow H\;\; \left|\;\;\int_Hf({\mathbf{q}})\overline{f({\mathbf{q}})}d\eta({\mathbf{q}})<\infty \right.\right\}.$$ This is a left quaternionic Hilbert space with a scalar product as in (\[left-sc-prod\]).

\[TH1\] The set $\left\{H_n({\mathbf{q}})\;\;\left| \;\; \right.n\in\mathbb{N}\right\}$ is an orthogonal set in $L_H^2(H,d\eta({\mathbf{q}}))$.

Consider $$\begin{aligned}
& &\int_HH_n({\mathbf{q}})\overline{H_m({\mathbf{q}})}d\eta({\mathbf{q}})\\
&=&\int_{SU(2)}u_{{\mathbf{q}}}\left(\begin{array}{cc}
\int_{{\mathbb{R}}^2}H_n(z)H_m(\overline{z})d\nu(x,y)&0\\0&\int_{{\mathbb{R}}^2}H_m(z)H_n(\overline{z})d\nu(x,y)\end{array}\right)u_{{\mathbf{q}}}^{\dagger}d\omega(u_{{\mathbf{q}}})\\
&=&\int_{SU(2)}u_{{\mathbf{q}}}\left(\begin{array}{cc}
b_n(s)n!\delta_{mn}&0\\
0&b_m(s)m!\delta_{mn}\end{array}\right)u_{{\mathbf{q}}}^{\dagger}d\omega(u_{{\mathbf{q}}})\\
&=&\int_{SU(2)}u_{{\mathbf{q}}}u_{{\mathbf{q}}}^{\dagger}d\omega(u_{{\mathbf{q}}})b_n(s)n!\delta_{mn}\\
&=&b_n(s)n!\delta_{nm}.\end{aligned}$$

Redefine $$\label{C3}
H_n^s ({\mathbf{q}})=b_n(s)^{-\frac{1}{2}}H_n({\mathbf{q}}),$$ then $H_n^s({\mathbf{q}})\in L_H^2(H,d\eta({\mathbf{q}}))$ and $$\label{H6}
\int_HH_n^s({\mathbf{q}})\overline{H_m^s({\mathbf{q}})}d\eta({\mathbf{q}})=\delta_{nm}.$$ Define 