of regular functions.

(Cullen derivative [@Gra1; @Gra2])\[D3\] Let $\Omega$ be a domain in $H$, and let $f:\Omega\longrightarrow H$ be a left regular function. The Cullen derivative of $f$, $\partial_cf$, is defined as $$\partial_c   f ({\mathbf{q}})=\left\{\begin{array}{ccc}
\partial_If({\mathbf{q}}):=\dfrac{1}{2}\left(\dfrac{\partial f_I(x+Iy)}{\partial x}-I\dfrac{\partial f_I(x+Iy)}{\partial y}\right)&\text{if}&y\not=0\\[4mm]
\dfrac{\partial f}{\partial x}(x)&\text{if}&{\mathbf{q}}=x~~\text{is real}\end{array}
\right.$$ Similarly, for a right regular function $f$ its Cullen derivative is defined as $$\partial_c   f ({\mathbf{q}})=\left\{\begin{array}{ccc}
\partial_If({\mathbf{q}}):=\dfrac{1}{2}\left(\dfrac{\partial f_I(x+Iy)}{\partial x}-\dfrac{\partial f_I(x+Iy)}{\partial y} I\right)&\text{if}&y\not=0\\[4mm]
\dfrac{\partial f}{\partial x}(x)&\text{if}&{\mathbf{q}}=x~~\text{is real}\end{array}
\right.$$

Under the above definition the Cullen derivative of a regular function is regular and with $a_n\in H$ we have, for example, for a right regular power series, $$\label{E11}
\partial_c\left(\sum_{n=0}^{\infty}a_n {\mathbf{q}}^n\right)=\sum_{n=0}^{\infty} n a_n {\mathbf{q}}^{n-1}.$$ The following theorem gives the quaternionic version of holomorphy via a Taylor series. Let $B(0,R)$ be an open ball in $H$, of radius $R$ and centered at $0$.

[@Gra1]\[T5\] A function $f: B(0,R)\longrightarrow H$ is right, respectively left, regular if and only if it has a series expansion of the form $$f({\mathbf{q}})=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\partial^n f}{\partial x^n}(0){\mathbf{q}}^n , \qquad\text{respectively,} \qquad f({\mathbf{q}})=\sum_{n=0}^{\infty}{\mathbf{q}}^n\frac{1}{n!}\frac{\partial^n f}{\partial x^n}(0),$$ converging on $B(0, R)$.

Complex Hermite polynomials
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For the construction and analysis of the quaternionic Hermite polynomials, $H_n({\mathbf{q}})$ and $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$, it would be useful to first review some facts about their complex counterparts, specially 