z\rangle=\mathcal N(|z|)^{-\frac{1}{2}}
\sum_{m=0}^{\infty}\frac{\Phi_{m}(z)}{\sqrt{\rho(m)}}
\phi_{m}\in\mathfrak{H};\;\;z\in{\mathfrak{D}},$$ where $\mathcal N(|z|)$ is a normalization factor and $\{\rho(m)\}_{m=0}^{\infty}$ is a sequence of nonzero positive real numbers. The set of vectors in (\[ma-cs\]) is said to form a set of CS if

1.  $\langle z\mid z\rangle=1$ for all $z\in{\mathfrak{D}}$;

2.  the states $\{\mid z\rangle \mid z\in{\mathfrak{D}}\}$ satisfy a resolution of the identity: $$\int_{\mathfrak D}\mid z\rangle \langle z\mid d\mu=I_{\mathfrak H}
    \label{2},$$

where $d\mu$ is an appropriately chosen measure and $I_{\mathfrak H}$ is the identity operator on $\mathfrak H$.

In [@Thi], we studied what were called quaternionic CS in a complex Hilbert space. These were defined as vector CS, built using a quaternionic variable. However, recently, in [@Thi2] we have defined the cannonical quaternionic CS by replacing the $z$ in (\[can-cs\]) by a quaternion ${\mathbf{q}}$ and considered these CS as vectors in a quaternionic Hilbert space (see also [@AdMil] for Prelemov type quaternionic CS). Further, we have also defined the quaternionic version of Hermite polynomials by replacing $z$ in $H_{n,m}(z,\overline{z})$ by a quaternion. Since there is more than one way of defining a derivative with respect to a quaternionic variable [@Am], in [@Thi2] the quaternionic derivative was introduced in a formal sense. However, recent developments of quaternionic analysis 