since the results we obtain here in the quaternionic case are parallel to those in the complex case. For a detailed discussion of the complex Hermite polynomials $H_n(z)$ and $H_{n,m}(z,\overline{z})$, we refer the reader to [@AFG; @Ga; @Ghan; @Mat; @Wil; @VE].

The polynomials $H_n(z)$
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Let $z=x+iy\in{\mathbb{C}}, \;\; 0<s<1$, $$d\nu(z)=d\nu(x,y)=\exp\left[-(1-s)x^2-(\frac{1}{s}-1)y^2\right]dxdy
\label{measure1}$$ and $$b_n(s)=\frac{\pi\sqrt{s}}{1-s}\left(2\frac{1+s}{1-s}\right)^n n!\, .
\label{normalization}$$ Define $$\label{H1}
H_n(z)=n!\sum_{m=0}^{[n/2]}\frac{(-1)^m(2z)^{n-2m}}{m!(n-2m)!}=n!\sum_{m=0}^{[n/2]}C_{nm}z^{n-2m}.$$ ($[x]$ denoting the integer part of $x$) and observe that $\overline{H_n(x+iy)}=H_n(x-iy)$. The $H_n$ are just the usual Hermite polynomials, written in terms of a complex argument. From [@Ga] and [@VE] we have $$\label{H2}
\int_{{\mathbb{R}}^2}H_n(x+iy)\overline{H_m(x+iy)}d\nu(x,y)=b_n(s)\; \delta_{nm}\;  .$$ Define $$\label{F3}
\mathfrak{h}_{n,s}(z)= b_n(s)^{-\frac{1}{2}}e^{-z^2/2}H_n(z)$$ and the Hilbert space of entire functions $$\mathfrak{X}_s=\left\{f~\mid~\int_{{\mathbb{R}}^2}|f(x+iy)|^2\exp(sx^2-\frac{1}{s}y^2)dxdy<\infty\right\}.$$ It has been shown in [@VE] that $\{\mathfrak{h}_{n,s}(z)\}_{n=0}^{\infty}$ is an orthonormal basis of $\mathfrak{X}_s$ and $\mathfrak{X}_s$ is a reproducing kernel Hilbert space with the kernel $$\label{F4}
\mathfrak{K}_s(z,\overline{w})=\sum_{n}\mathfrak{h}_{n,s}(z)\overline{\mathfrak{h}_{n,s}(w)}=\frac{1-s^2}{2\pi s}\exp\left[-\frac{1+s^2}{4s}(z^2+\overline{w}^2)+\frac{1-s^2}{2s}z\overline{w}\right],$$ where $\quad z,w\in{\mathbb{C}}$. The expression for the kernel can be reduced to (see [@Ga]) $$\label{F5}
\mathfrak{K}_s(z,\overline{z})=\frac{1-s^2}{2\pi s}\exp[-sx^2+\frac{1}{s}y^2],\quad z=x+iy.$$ If we take $$\label{H3}
h_{n,s}(z)= b_n(s)^{-\frac{1}{2}}H_n(z)$$ then for $z,w\in\mathbb{C}$, $$\label{H4}
K_s(z,\overline{w})=\sum_{n}h_{n,s}(z)\overline{h_{n,s}(w)}=\frac{1-s^2}{2\pi s}\exp\left[-\frac{(s-1)^2}{4s}(z^2+\overline{w}^2)+\frac{1-s^2}{2s}z\overline{w}\right].$$ Further by replacing $w$ by $z$ we get $$\label{H5}
K_s(z,\overline{z})=\frac{1-s^2}{2\pi s}\exp\left[\frac{1-s}{2}x^2+\frac{s^2-3s+2}{2s}y^2\right],\quad z=x+iy.$$ The kernel (\[H4\]) is also a reproducing kernel. In fact it is easily seen that the corresponding reproducing kernel Hilbert space, $\mathfrak H^s_{hol}$, 