  Starting from the pseudo-differential decomposition
$\mathbf{D}=(-\Delta)^{\frac{1}{2}}\mathcal{H}$ of the Dirac operator
$\displaystyle \mathbf{D}=\sum_{j=1}^n\mathbf{e}_j\partial_{x_j}$ in terms of
the fractional operator $(-\Delta)^{\frac{1}{2}}$ of order $1$ and of the
Riesz-Hilbert type operator $\mathcal{H}$ we will investigate the fundamental
solutions of the space-fractional Dirac equation of L\'evy-Feller type
$$\partial_t\Phi_\alpha(\mathbf{x},t;\theta)=-(-\Delta)^{\frac{\alpha}{2}}\exp\left(
\frac{i\pi\theta}{2} \mathcal{H}\right)\Phi_\alpha(\mathbb{x},t;\theta) $$
involving the fractional Laplacian $-(-\Delta)^{\frac{\alpha}{2}}$ of order
$\alpha$, with $2m\leq \alpha <2m+2$ ($m\in \mathbb{N}$), and the
exponentiation operator $\exp\left( \frac{i\pi\theta}{2} \mathcal{H}\right)$ as
the hypercomplex counterpart of the fractional Riesz-Hilbert transform carrying
the \textit{skewness parameter} $\theta$, with values in the range
$|\theta|\leq \min\{\alpha-2m,2m+2-\alpha\}$. Such model problem permits us to
obtain hypercomplex counterparts for the fundamental solutions of higher-order
heat-type equations $\partial_t F_M(x,t)=\kappa_M(\partial_x)^M F_M(x,t)$
$(M=2,3,\ldots)$ in case where the even powers resp. odd powers
$\mathbf{D}^{2m}=(-\Delta)^{m}$ ($M=\alpha=2m$) resp.
$\mathbf{D}^{2m+1}=(-\Delta)^{m+\frac{1}{2}}\mathcal{H}$ ($M=\alpha=2m+1$) of
$\mathbf{D}$ are being considered.
