text:
\section{Introduction}

\subsection{The State of Art}
The study of null solutions of Dirac-like operators is known as the heart of several function theories in the context of Clifford algebras. From the knowledge that Dirac-like operators factorize the Laplace operator and its analogues, it is desired to study it algebraically and analytically in order to obtain refinements of well known results in harmonic analysis and to develop applications in the fields of mathematical physics and applied mathematics as well.

In 1967 L\'evy-Leblond investigates the factorization of the Schr\"odinger operator in order to produce non-relativistic analogues of the Dirac operator in the $1+3$--dimensional space carrying any spin (cf.~\cite{LevyLeblond67}). However we have to wait for contemporary times to see how the $(1+n)-$dimensional generalization of the L\'evy-Leblond type picture, coined as parabolic Dirac type operators, can be used to cover a wide range of applications on the crossroads of function theory and boundary value problems. Between the many notable results regarding this new class of operators, it is worth to stress the importance of the groundbreaking works of Cerejeiras, K\"ahler \& Sommen \cite{CKS05} and Cerejeiras, Sommen \& Vieira \cite{CSV07} who proposed the matter of factorize the heat operator and its relatives. Since then, their work motivated many researchers to exploit this framework to several contexts, including higher-dimensional analogues of nonlinear Schr\"odinger equations \cite{B06,CFV08}.

In the last decade the L\'evy-Leblond picture had continue to receiving an increasing interest on the formulation of several physical-phase-space formulations of spinning particles through supersymmetric Lie algebraic representations of PDEs (see \cite{AKTT16,A18}) as well as on operational models involving parabolic Dirac operators and their fractional analogues (see e.g. \cite{FV16,FV17,BCBM20,BRS20} and the references therein), to mention a few. Therewith, it is reasonable to say the L\'evy-Leblond picture is nowadays well understood.

In parallel, the study of discrete Dirac operators has experienced a rapid increase in hypercomplex analysis during the last fithteen years, strongly influenced by the pioneering papers \cite{FK07,FKS07,BSSV08,RSKS10}.
The literature towards this topic, whose physical roots may be found e.g. on the research papers of Kogut \& Susskind \cite{KS75} and Rabin \cite{Rabin82} (see also \cite{KK04,Sushch14} and the references therein) is very diverse. To the purpose of this paper we will depict a short overview to motivate our approach.

For the construction of a faithful discretization of the discrete Dirac operator on the lattice $h{\mathbb Z}^n$, say  $D_h$, whose precise definition will be introduced afterwards, the Clifford algebra of signature of $(n,n)$ may be embody on the ladder structure of $D_h$ with the aid of the so-called Witt basis $\{{\bf e}_j^+,{\bf e}_j^-~:~j=1,2,\ldots,n\}$ to resemble the canonical structure of the exterior algebra (cf.~\cite[Chapter 2]{VR16}).

 Originally highlighted in \cite{FKS07} and on the series of papers \cite{BSSV08,RSKS10}, such fact was only clarified recently in author's recent paper \cite{FaustinoKGordonDirac16}, through the canonical isomorphism $C \kern -0.1em \ell_{n,n}\cong \mbox{End}(C \kern -0.1em \ell_{0,n})$ between the Clifford algebra $C \kern -0.1em \ell_{n,n}$ of signature $(n,n)$ and the algebra of endomorphisms acting on the Clifford algebra $C \kern -0.1em \ell_{0,n}$ of signature $(0,n)$.

This in turn allows us to establish a canonical correspondence between the discrete counterpart of the Dirac-K\"ahler operator $d-d^*$ and the multivector approximation $D_h$ of Dirac operator over $h{\mathbb Z}^n$ (see also \cite{FaustinoMMAS17}).

In a larger extend,
starting from forward/backward discretizations $D_h^\pm$
of the Dirac operator $\displaystyle D=\sum_{j=1}^{n}{\bf e}_j\partial_{x_j}$ formely introduced in \cite{FK07}, one can make use of the wedge ($\wedge$) and the dot $(\bullet)$ actions on $C \kern -0.1em \ell_{0,n}$ to establish the canonical one-to-one correspondences
\begin{eqnarray*}
	d\overset{\text{1-1}} \longleftrightarrow D_h^- \wedge (\cdot) & \mbox{and}& d^* \overset{\text{1-1}} \longleftrightarrow D_h^+ \bullet (\cdot)
\end{eqnarray*}
in a way that the discrete Dirac operator $D_{h}
$ yields as a $C \kern -0.1em \ell_{n,n}-$valued representation for the multivector counterpart $$D_h^- \wedge (\cdot) -D_h^+ \bullet (\cdot)$$
of the Dirac-K\"ahler operator (cf.~\cite[subsection 2.3]{FaustinoKGordonDirac16}).

\newpage
\subsection{Main targets}

\begin{figure}\label{LevyLeblondPicture}
	\begin{tikzpicture}
		\node[mynode] (v1){\textbf{Time-fractional regularization}\\ \ \\ $	D_h-{\mathfrak f} e^{i\theta(1-\beta)} \mathbb{D}_t^\beta +{\mathfrak f}^\dagger e^{-i\theta}$};
		\node[mynode,below right=3.0cm of v1](v2) {
			\textbf{Semidiscrete Dirac operator} \\ \ \\ $\displaystyle D_h+{\mathfrak f} \partial_t +{\mathfrak f}^\dagger e^{-i\theta} $};

		\node[mynode,below = 4.5cm of v1] (v4){\textbf{Space-fractional regularization}  \\ \ \\   $D_h+ {\mathfrak f}\partial_t(-\Delta_h)^{1-\alpha} +e^{-i\theta} {\mathfrak f}^\dagger$};
		\node[mynode,below=2.7cm of v2](v3) {{\bf L\'evy-Leblond type operator} \\ \ \\
			$D+{\mathfrak f}\partial_t+{\mathfrak f}^\dagger e^{-i\theta}$ \\ \ \\ (cf.~\cite{CKS05,B06,CSV07,BCBM20})};
		\node[mynode, above=4.0cm of v2](v5) {{\bf Discrete Dirac Operator} \\ \ \\
			$D_h\overset{\text{1-1}} \longleftrightarrow D_h^-\wedge (\cdot)-D_h^+\bullet (\cdot)$  \\ \ \\
			(cf.~\cite{FaustinoKGordonDirac16,FaustinoMMAS17,F19b})};

		\draw[-latex] (v2.north) -- node[auto,] {$\beta \geq 1$} (v1.east);
		\draw[-latex] (v1.south) -- node[auto,] {$\beta=\dfrac{1}{\alpha}$} (v4.north);
		\draw[-latex] (v4.north) -- node[auto,] {$|\theta|\leq \dfrac{\alpha\pi}{2}$} (v1.south);

		\draw[-latex] (v2.south) -- node[below=3mm, align=center] {$0<\alpha\leq 1$} (v4.east);

		\draw[-latex] (v5.south) -- node[auto,]{Witt basis ${\mathfrak f},{\mathfrak f}^\dagger$} (v2.north);
		\draw[-latex] (v2.south) -- node[auto,]{$h\rightarrow 0$} (v3.north);
	\end{tikzpicture}
	\caption{The L\'evy-Leblond picture on the fractional semidiscrete case.}
\end{figure}
In this paper we will focus our attention
on the time-fractional and space-fractional Dirac operators
carrying the parameters $0<\alpha\leq 1$, $\beta\geq 1$ and $|\theta|\leq\frac{\alpha \pi}{2}$.

On the construction neatly summarized in {\sc Figure 1.} the notation $\Delta_h$ stands for the discrete Laplacian on the lattice $h{\mathbb Z}^n$ considered in several author's contributions such as \cite{FaustinoKGordonDirac16,FaustinoMMAS17}. In addition, the family of fractional discrete operators $(-\Delta_h)^{\sigma}$ ($0<\sigma\leq 1$) will be defined in terms of its Fourier symbol in the streamlines of \cite[Section 6.]{LizamaRoncal18} (see also \cite[section 21.4.3]{F19}). For our main purposes we will adopt the notation $\mathbb{D}_{t}^\beta$ for the so-called Riemann-Liouville (right-sided) fractional derivative (cf.~\cite[Chapter 2]{SKM93}).

The operator $\mathbb{D}_{t}^\beta$ defined as follows:
\begin{eqnarray}
	\label{RiemannLiouville}
\mathbb{D}_{t}^\beta\Psi(x,t)=	\left\{\begin{array}{lll}
		\displaystyle 	(-\partial_t)^k \int_{t}^{+\infty}g_{k-\beta}(s-t)\Psi(x,s)ds & \mbox{for} & k-1<\beta<k
		\\ \ \\
		(-\partial_t)^k \Psi(x,t)& \mbox{for} & \beta=k,
	\end{array}\right.
\end{eqnarray}
where $k=\lfloor \beta \rfloor+1$ ($\lfloor \beta \rfloor$ denotes the integer part of $\beta$), is an integro-differential operator for every $\beta\neq k$, involving the higher order time-derivative $(-\partial_t)^k:=(-1)^k(\partial_t)^k$, and an integral part corresponding to the convolution between $\Psi(x,t)$ and
the Gel'fand-Shilov function $g_{\nu}:\mathbb{R}\rightarrow [0,\infty)$, defined for $-\nu \not \in {\mathbb N}_0$ by
	\begin{eqnarray}
	\label{GelfandShilov}g_\nu(p)=\left\{\begin{array}{lll}
 \displaystyle 	\frac{p^{\nu-1}}{\Gamma(\nu)} & \mbox{for} & p>0
		\\ \ \\
0 & \mbox{for} &p\leq 0.
	\end{array}\right.
\end{eqnarray}

It should be noted that the Eulerian integral representation involving the Gamma function $\Gamma(\nu)$ (cf.~\cite[formula 2.3.3. (1), p.322]{PBM86})
\begin{eqnarray}
\label{LaplaceId}\int_0^\infty e^{-p\lambda} p^{\nu-1} dp=\Gamma(\nu)\lambda^{-\nu}, & \Re(\nu)>0~~\&&~~  \Re(\lambda)>0
\end{eqnarray}
assures that, for $k-1<\beta <k$, the function $p\mapsto g_{k-\beta}(p)$ appearing on the definition of $\mathbb{D}_{t}^\beta$ defines a probability density function, converging to the Dirac delta function, that is
$$
\lim_{\beta \rightarrow k^-}g_{k-\beta}(p)=\delta(p).
$$

The later properties allows us to say that $\mathbb{D}_{t}^\beta$ ($k-1<\beta <k$) provides us a regularization of $(-\partial_t)^k$ in the sense of the topology of the underlying space of tempered distributions.

The motivation to this paper comes from the series of contributions \cite{CiaurriGRTV18,LizamaRoncal18,GLizamaM21} focused on the study of Cauchy problems involving fractional discrete Laplacians, and from the interrelationship between space-fractional and time-fractional operators highlighted in \cite{CLRV15} towards superdiffusion equations. Such link has played a crucial role in the theory of PDEs (cf.~\cite{KL02,KL12}) and on stochastic calculus as well (cf.~\cite{BMN09}). For a survey on both topics, we also refer to the monograph \cite{MS12}.

Thus, we are not only interested fractional difference analogues for the parabolic operator of heat type ($\alpha=\beta=1$, $\theta=0$ \& $h\rightarrow 0$) and of Schr\"odinger type ($\alpha=\beta=1$, $\theta=\pm\frac{\pi}{2}$ \& $h\rightarrow 0$) considered in \cite{CKS05,B06,CSV07,AKTT16,A18,BCBM20}, neither fractional counterparts of semidiscrete models involving the {\it semidiscrete heat operator} $\partial_t-\Delta_h=(D_h+{\mathfrak f}\partial_t+{\mathfrak f}^\dagger)^2$ (i.e when $\alpha=\beta=1$ \& $\theta=0$), already considered in~\cite{BBRS14},
but also on the interface between space-fractional and time-fractional in a way that the {\it fractional discrete Laplacian} $-(-\Delta_h)^\alpha$ carrying the parameter $0<\alpha\leq 1$ turns out be treated as a temporal regularization of order $\beta:=\frac{1}{\alpha}\geq 1$ encoded on the time-fractional derivative $\mathbb{D}_t^\beta$.

Moreover, the parameter condition $|\theta|\leq \frac{\alpha\pi}{2}$ encoded on the [fractional semidiscrete] analytic semigroup $\left\{\exp(-te^{i\theta}(-\Delta_h)^\alpha)\right\}_{t\geq 0}$ is ubiquitous on space-fractional diffusion models (cf.~\cite{Mainardi01}). Its incorporation on our model problem is seamlessly justified by the constraint $|\theta|=|\arg(te^{i\theta})|<\pi$ carrying the integral representation obtained in \cite[p. 458, eq.~(21.35)]{F19} for {\it modified Bessel functions of the first kind} (cf.~\cite[subsection 2.1 \& subsection 3.1]{LizamaRoncal18}). This will be the main novelty of this paper in comparison with the semidiscrete heat semigroup representations considered in the series of author's contributions \cite{F19,F19b,F20}.

Up to author's knowledge, the overlap between space-fractional Dirac operators -- such as the ones introduced by Bernstein in \cite{B16} -- and time-fractional Dirac operators of Riemann-Liouville type -- such as the ones considered e.g. by Ferreira \& Vieira \cite{FV16} --  was not yet addressed so that the idea of connecting fractional order in time and space on this paper shall be seen eventually as another step further to improve e.g. the study of mild solutions for time-fractional Navier-Stokes equations in the superdiffusive case, from a hypercomplex analysis perspective (see e.g. \cite{CarvalhoPlanas15,ZP17} and the references therein for an overview on the subdiffusive case).

\subsection{Layout of the paper}
This paper is organized as follows:
\begin{itemize}
\item In section \ref{DefsSection} we briefly provide some background on Clifford algebras and on {\it discrete Fourier analysis} required throughout the paper. From the point of view of the theory of pseudo-differential operators, that will allows us sift all the well-known constructions in the space of square-integrable functions to the space of Clifford-valued distributions over the lattice $h{\mathbb Z}^n$ (see, for instance, \cite[subsection 21.2]{F19} \& \cite[subsection 2.2.]{F19}).
\item In section \ref{SemidiscreteSection} we introduce a time-fractional and a space-fractional variant of the {\it semidiscrete Dirac operator} $D_h+{\mathfrak f}\partial_t+{\mathfrak f}^\dagger e^{-i\theta}$.
Guided by the approaches of Cerejeiras, K\"ahler \& Sommen \cite{CKS05} and Cerejeiras, Sommen \& Vieira \cite{CSV07}, we introduce the time-fractional regularization of $D_h+{\mathfrak f}\partial_t+{\mathfrak f}^\dagger e^{-i\theta}$ by replacing the time-derivative $\partial_t$ by a time-fractional counterpart $-e^{i\theta(1-\beta)}\mathbb{D}_t^\beta$ ($\beta\geq 1$) mixing the Riemann-Liouville derivative and a unitary term lying on the unit circle $\mathbb{S}^1$. For the space-fractional regularization, we consider the space-fractional counterpart $\partial_t(-\Delta_h)^{\alpha-1}$ of $\partial_t$, involving the fractional discrete operator $(-\Delta_h)^{\alpha-1}$ ($0<\alpha\leq 1$) instead of $\partial_t$ (case of $\alpha=1$). With the proof of {\bf Theorem \ref{CoupledSystemParabolicDiracRLt}} \& {\bf Theorem \ref{CoupledSystemParabolicDiract}} we will show that the formulations highlighted on {\sc Figure 1} retain all of the salient features of the null solutions of the {\it L\'evy-Leblond type operator} $D+{\mathfrak f} \partial_t+{\mathfrak f}^\dagger e^{-i\theta}$ (limit case $h\rightarrow 0$) considered by several authors (see also the papers of Bernstein \cite{B06}, and Bao, Constales, De Bie \& Mertens \cite{BCBM20}).
\item In section \ref{MainResultsSection} we will show that the null solutions of both fractional semidiscrete operators are indeed interrelated.
 Starting from the pseudo-differential representation of the fractional semidiscrete analytic semigroup $\{\exp(-te^{i\theta}(-\Delta_h)^\alpha)\}_{t\geq 0}$ in terms of its Fourier symbol, we will provide first, with the proof of {\bf Theorem \ref{SemigroupCauchySolver}}, a bridge result that establishes the correspondence between Cauchy problems of space-fractional and time-fractional type encoded by the set of fractional operators, $e^{-i\theta}\partial_t+(-\Delta_h)^\alpha$ and $-e^{-i\theta}\mathbb{D}_t^\beta-\Delta_h$, respectively. Such result is essentially a wise reformulation of \cite[Theorem 3]{CLRV15} in terms of Riemann-Liouville (right-sided) fractional derivative.
Afterwards, we will prove {\bf Theorem \ref{LevyLeblondSolutions}} -- the main contribution of this paper -- by combining {\bf Theorem \ref{CoupledSystemParabolicDiracRLt}}, {\bf Theorem \ref{CoupledSystemParabolicDiract}} \& {\bf Theorem \ref{SemigroupCauchySolver}}.
\item Finally, in section \ref{SpaceFractionalRemarks}, we will briefly discuss an alternative formulation for the {\it semidiscrete Dirac operator of space-fractional type} that factorizes the space-fractional operator $e^{-i\theta}\partial_t+(-\Delta_h)^\alpha$.
\end{itemize}

\section{Definitions}\label{DefsSection}

\subsection{Clifford algebra setup}

Let ${\bf e}_0,{\bf e}_1,{\bf e}_2,\ldots,{\bf e}_n,{\bf e}_{n+1},{\bf e}_{n+2}\,\ldots,{\bf e}_{2n},{\bf e}_{2n+1}$ be the generators of the Clifford algebra $C \kern -0.1em \ell_{n+1,n+1}$ of signature $(n+1,n+1)$, satisfying
\begin{eqnarray}
	\label{CliffordBasis}
	\begin{array}{lll}
		{\bf e}_j {\bf e}_k+ {\bf e}_k {\bf e}_j=-2\delta_{jk}, & 0\leq j,k\leq n \\
		{\bf e}_{j} {\bf e}_{n+k}+ {\bf e}_{n+k} {\bf e}_{j}=0, & 0\leq j\leq n ~~\&~~  1\leq k\leq n+1\\
		{\bf e}_{n+j} {\bf e}_{n+k}+ {\bf e}_{n+k} {\bf e}_{n+j}=2\delta_{jk}, & 1\leq j,k\leq
		n+1.
	\end{array}
\end{eqnarray}

As it is well-known from the literature (cf.
\cite[Chapter 3]{VR16}), the Clifford algebra $C \kern -0.1em \ell_{n+1,n+1}$ is a
universal algebra of dimension $2^{2n+2}$, linear isomorphic to the
exterior algebra $\Lambda^*\left({\mathbb R}^{n+1,n+1}\right)$  containing the field of real numbers ${\mathbb R}$, the $n-$dimensional Euclidean space ${\mathbb R}^n$, and the Minkowski space ${\mathbb R}^{n+1,n+1}$ of signature $(n+1,n+1)$ as proper subspaces.

In particular, the ladder structure of $C \kern -0.1em \ell_{n+1,n+1}$ allows us to represent the space-time tuple $(x_1,x_2,\ldots,x_n,t)$ of ${\mathbb R}^{n+1}$ through the paravector representation $\displaystyle t+x=t+\sum_{j=1}^{n}x_j{\bf e}_j$ of  ${\mathbb R}\oplus {\mathbb R}^n$.

Here and elsewhere, the $1-$vector
representations $\displaystyle x=\sum_{j=1}^{n}x_j{\bf e}_j$ and $\displaystyle x\pm h
{\bf e}_j$ of ${\mathbb R}^{n}$ will be used to describe
the lattice point $(x_1,x_2,\ldots,x_n)$ of $h{\mathbb Z}^{n}$, and the
forward/backward shifts $(x_1,x_2,\ldots,x_j\pm h,\ldots, x_n)$ over
$h{\mathbb Z}^n$, respectively.
For a sake of readibility , one will use throughout the manuscript the tuple notations $(x+h{\bf e}_j,t)$ and $(x-h{\bf e}_j,t)$
to define shifts over the semidiscrete space-time lattice
$$h{\mathbb Z}^n \times \left[0,\infty\right):=\left\{(x,t)\in {\mathbb R}^{n+1}~:~\frac{x}{h}\in {\mathbb Z}^n~~\wedge~~t\geq 0 \right\}.$$

We notice also that the Clifford algebras $C \kern -0.1em \ell_{0,n}$, $C \kern -0.1em \ell_{1,1}$ and $C \kern -0.1em \ell_{1,n+1}$ considered e.g. on the series of papers \cite{CKS05,B06,FV16,FV17,BCBM20} are subalgebras of $C \kern -0.1em \ell_{n+1,n+1}$. For our purposes we assume that $C \kern -0.1em \ell_{1,1}$ is a Clifford algebra generated by the nilpotents
\begin{eqnarray}
\label{WittBasis}	{\mathfrak f}=\frac{1}{2}({\bf e}_{2n+1}+{\bf e}_0) & \mbox{and} & {\mathfrak f}^\dagger=\frac{1}{2}({\bf e}_{2n+1}-{\bf e}_0).
\end{eqnarray}

That is,
\begin{eqnarray}
		\label{CliffordWittBasis}
	\begin{array}{lll}
		{\bf e}_j  {\mathfrak f}+{\mathfrak f} {\bf e}_j=0, & {\bf e}_{n+j} {\mathfrak f}+{\mathfrak f} {\bf e}_{n+j}=0, &(1\leq j\leq n)\\ \ \\
	{\bf e}_j  {\mathfrak f}^\dagger+{\mathfrak f}^\dagger {\bf e}_j=0, & {\bf e}_{n+j} {\mathfrak f}^\dagger +{\mathfrak f}^\dagger {\bf e}_{n+j}=0, &(1\leq j\leq n)\\ \ \\
	({\mathfrak f})^2=({\mathfrak f}^\dagger)^2=0,& {\mathfrak f} {\mathfrak f}^\dagger +{\mathfrak f}^\dagger {\mathfrak f}=1. &
	\end{array}
\end{eqnarray}

Furthermore, based on decomposition $C \kern -0.1em \ell_{n+1,n+1}=C \kern -0.1em \ell_{1,1}\otimes C \kern -0.1em \ell_{n,n}$, we will represent the Clifford-vector-valued functions $(x,t)\mapsto \Psi(x,t)$ with membership in the {\it complexified Clifford algebra} ${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}$ through the ansatz
\begin{eqnarray*}
	\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak f}\Psi^{[1]}(x,t)+{\mathfrak f}^\dagger \Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger \Psi^{[3]}(x,t),
\end{eqnarray*}
whereby $(x,t)\mapsto\Psi^{[m]}(x,t)$ ($m=0,1,2,3$) are Clifford-vector-valued functions with membership in ${\mathbb C} \otimes C \kern -0.1em \ell_{n,n}$.

To introduce in the sequel function spaces and operators underlying to the ${\mathbb C} \otimes C \kern -0.1em \ell_{n+1,n+1}-$valued functions $(x,t)\mapsto \Psi(x,t)$ and its ${\mathbb C}\otimesC \kern -0.1em \ell_{n,n}-$valued components $(x,t)\mapsto\Psi^{[m]}(x,t)$ ($m=0,1,2,3$), as well as for the ${\mathbb C}\otimesC \kern -0.1em \ell_{n,n}-$valued representations of the discrete Dirac operator $D_h$ in terms of its Fourier multipliers (cf.~\cite[subsection 21.2.2]{F19} and \cite[subsection 2.3]{F20}), one
has to consider the $\dag-${\it
 	conjugation} operation ${\bf a} \mapsto{\bf a}^\dag$ on the \textit{complexified Clifford algebra} ${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}$, defined recursively as follows:
 \begin{eqnarray}
 	\label{dagconjugation}
 	\begin{array}{lll}
 		(\a \b)^\dag=\b^\dag\a^\dag \\ (a {\bf e}_J)^\dag =\overline{a_J}~{\bf e}_{j_r}^\dag
 		\ldots {\bf e}_{j_2}^\dag{\bf e}_{j_1}^\dag~~~(0\leq j_1<j_2<\ldots<j_r\leq 2n+1) \\
 		{\bf e}_j^\dag=-{\bf e}_j~~~\mbox{and}~~~{\bf e}_{n+1+j}^\dag={\bf e}_{n+1+j}~~~(0\leq j\leq
 		n).
 	\end{array}.
 \end{eqnarray}

From (\ref{dagconjugation}), the $\dagger-$conjugation identities
\begin{center}
$({\mathfrak f}^\dag)^\dag={\mathfrak f}$ and $({\mathfrak f} {\mathfrak f}^\dagger)^\dagger={\mathfrak f} {\mathfrak f}^\dagger$
\end{center}
involving the nilpotents ${\mathfrak f}$ and ${\mathfrak f}^\dagger$ defined viz eq. (\ref{WittBasis}) are then immediate. Also, from (\ref{dagconjugation}) one readily has that $$\a\mapsto \| \a\|:=\sqrt{\a^\dagger \a}$$
defines a $\|\cdot \|-$norm endowed by \textit{complexified Clifford algebra} structure ${\mathbb C}\otimes C \kern -0.1em \ell_{n+1,n+1}$, since $\a^\dag\a=\a\a^\dag$ is a non-negative real number. In case where $\a$ belongs to ${\mathbb C}\otimes {\mathbb R}^{n+1,n+1}$, the quantity $\|\a\|$ equals to the standard norm of $\a$ on ${\mathbb C}^{2n+2}$.

\subsection{The discrete Fourier analysis background}

Let us define by $\mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n}):=\mathcal{S}(h{\mathbb Z}^n)\otimes \left({\mathbb C}\otimes C \kern -0.1em \ell_{n,n}\right)$ the Schwartz class of ${\mathbb C}\otimesC \kern -0.1em \ell_{n,n}-$valued functions on the lattice $h{\mathbb Z}^n$, consisting on \textit{rapidly decaying functions} $x\mapsto \varPhi(x,t)$ ($t \in [0,\infty)$) defined for any $M\in [0,\infty)$ by the semi-norm condition
$$ \displaystyle \sup_{x \in h{\mathbb Z}^n} (1+\| x\|^2)^M~\| \varPhi(x,t)\|<\infty,$$
and by $\ell_2(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n}):=\ell_2(h{\mathbb Z}^n)\otimes \left({\mathbb C}\otimes C \kern -0.1em \ell_{n,n}\right)$ the \textit{right Hilbert module} endowed by the Clifford-valued sesquilinear form
\begin{eqnarray}
	\label{lpInner} \langle \varPhi(\cdot,t),\varPsi(\cdot,t) \rangle_{h}=\sum_{x\in h{\mathbb Z}^n}h^n~
	\varPhi(x,t)^\dag\varPsi(x,t).
\end{eqnarray}

By exploiting \cite[Exercise 3.1.7]{RuzhanskyT10} to the Clifford-valued setting, it is easy to check that the seminorm condition
$$ \displaystyle \sup_{x \in h{\mathbb Z}^n} (1+\| x\|^2)^{-M}~\| \varPhi(x,t)\|<\infty$$
induces the set of all \textit{continuous linear functionals} with membership in the Schwarz class $\mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$ induced by the mapping $$\varPhi(\cdot,t) \mapsto \langle \varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h},$$ whereby the family of functions $\varPsi(\cdot,t):h{\mathbb Z}^n \rightarrow {\mathbb C}\otimes C \kern -0.1em \ell_{n,n}$ ($t\in [0,\infty)$) belong to the space of ${\mathbb C}\otimes C \kern -0.1em \ell_{n,n}-$valued tempered distributions on the lattice $h{\mathbb Z}^n$, that will be denoted by $$\mathcal{S}'(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n}):=\mathcal{S}'(h{\mathbb Z}^n)\otimes\left({\mathbb C}\otimes C \kern -0.1em \ell_{n,n}\right).$$

In particular, the mapping property $\varPhi(\cdot,t) \mapsto \langle \varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h}$ together with density arguments allows us to define, for every $x\mapsto \varPhi(x,t)$ with membership in
$
\ell_2(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})
$, a distribution $\varPhi(\cdot,t) \mapsto \langle \varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h}$ lying to $\mathcal{S}'(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$.

Next, we denote by $\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n$ the
$n-$dimensional \textit{Brioullin zone} representation for the $n-$torus ${\mathbb R}^n/\frac{2\pi}{h}{\mathbb Z}^n$ (cf.~\cite[p.~
324]{Rabin82}), by $$L_2\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C}\otimesC \kern -0.1em \ell_{n,n}\right):=L_2\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n\right)\otimes\left({\mathbb C}\otimes C \kern -0.1em \ell_{n,n}\right)$$ the ${\mathbb C} \otimes C \kern -0.1em \ell_{n,n}-$\textit{Hilbert module} endowed by the sesquilinear form
\begin{eqnarray}
	\label{BilinearFormQh}\langle {\bf f}(\cdot,t),\textbf{g}(\cdot,t
	)\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n}= \int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n} {\bf f}(\xi,t)^\dag \textbf{g}(\xi,t) d\xi,
\end{eqnarray}
and by $C^\infty(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$ the space of ${\mathbb C} \otimes C \kern -0.1em \ell_{n,n}-$valued test functions.
The \textit{discrete Fourier transform} defined by
\begin{eqnarray}
	\label{discreteFh}
	(\mathcal{F}_{h} \varPhi)(\xi,t)=\left\{\begin{array}{lll}
		\displaystyle \frac{h^n}{\left(2\pi\right)^{\frac{n}{2}}}\displaystyle
		\sum_{x\in h{\mathbb Z}^n}\varPhi(x,t)e^{i x \cdot \xi}, & \xi\in \left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n
		\\ \ \\
		0 , & \xi\in {\mathbb R}^n \setminus \left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n
	\end{array}\right.
\end{eqnarray}
 yields the isometric isomorphism $$\mathcal{F}_{h}:\ell_2(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})\rightarrow L_2\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}\right),$$ whose inverse $(\mathcal{F}_{h}^{-1} \textbf{g})(x,t)=\widehat{\textbf{g}}_{h}(x,t)
$ is provided by the Fourier coefficients
\begin{eqnarray}
	\label{FourierInversion}
	\widehat{\textbf{g}}_{h}(x,t)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n} (\mathcal{F}_{h} \textbf{g})(\xi,t) e^{-i x \cdot \xi} d\xi.
\end{eqnarray}

Moreover, by noticicing that the sesquilinear form (\ref{BilinearFormQh}) allows us to define a mapping that identifies $C^\infty(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$ with the dual space $C^\infty(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})'$, the so-called space of ${\mathbb C}\otimes C \kern -0.1em \ell_{n,n}-$valued distributions over $\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n$ (cf.~\cite[Exercise 3.1.15.]{RuzhanskyT10} \&  \cite[Definition 3.1.25]{RuzhanskyT10}),
we immediately get that the \textit{Parseval type relation}
\begin{eqnarray}
	\label{Parseval}	\langle \mathcal{F}_{h} \varPhi(\cdot,t),\textbf{g}(\cdot,t
	)\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n}=\left\langle \varPhi(\cdot,t),\widehat{\textbf{g}}_{h}(\cdot,t)\right\rangle_{h}
\end{eqnarray}
 involving the sesquilinear forms (\ref{lpInner}) and (\ref{BilinearFormQh}) (cf.~\cite[Definition 3.1.27]{RuzhanskyT10}) allows us to extend propertly (\ref{discreteFh}) to the setting of distributions, through the mapping property (cf.~\cite[Definition 3.1.27 \& 3.1.28]{RuzhanskyT10})
$$\mathcal{F}_{h}:\mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})\rightarrow C^\infty\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}\right)$$
in a way that the Fourier coefficients $x\mapsto \widehat{\textbf{g}}_{h}(x,t)$ defined {\it viz} eq. (\ref{FourierInversion}) belong to $\mathcal{S}'(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$.

In the same order of ideas of \cite[subsection 21.1.3]{F19} \& \cite[subsection 2.2]{F20} (see also \cite[section 6]{CiaurriGRTV17}), one can also define the \textit{discrete convolution operation} $\star_{h}$ between the \textit{discrete distribution} $\varPsi(\cdot,t)\in \mathcal{S}'(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$, and the \textit{discrete function} $\Phi(x)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$ as follows:
\begin{eqnarray}
	\label{discreteConvolution} \left(\varPsi(\cdot,t)\star_{h} \Phi\right)(x)=\sum_{y\in h{\mathbb Z}^n} h^n \Phi(y)\varPsi(y-x,t).
\end{eqnarray}

Indeed, from the duality condition
\begin{eqnarray*}
	\left\langle~ \varPsi(\cdot,t) \star_{h} \Phi,\textbf{g}(\cdot,t)~\right\rangle_{h}=\langle~ \varPsi(\cdot,t),\widetilde{\Phi} \star_{h} \textbf{g}(\cdot,t)~\rangle_{h}, & \mbox{with}& \widetilde{\Phi}(x)=[\Phi(-x)]^\dagger
\end{eqnarray*}
that yields straightforwardly from the sequence of identities
\begin{eqnarray*}
	\langle~\mathcal{F}_{h}\left[\varPsi(\cdot,t)\star_{h} \Phi\right],\textbf{g}(\cdot,t) ~\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n}&=&
	\langle~\varPsi(\cdot,t)\star_{h} \Phi, \mathcal{F}_{h}^{-1}[\textbf{g}(\cdot,t)]~ \rangle_{h}
	\\
	&=&\langle~\varPsi(\cdot,t), \widetilde{\Phi}\star_{h}\mathcal{F}_{h}^{-1}[\textbf{g}(\cdot,t)]~ \rangle_{h} \\
	&=&\left\langle~\varPsi(\cdot,t),\mathcal{F}_{h}^{-1}\left( \mathcal{F}_{h}\widetilde{\Phi}~\textbf{g}(\cdot,t)\right)~ \right\rangle_{h} \\
	&=&\left\langle~\mathcal{F}_{h}\varPsi(\cdot,t), \mathcal{F}_{h}\widetilde{\Phi}~\textbf{g}(\cdot,t)~ \right\rangle_{h} \\
	&=&\langle~\left(\mathcal{F}_{h}\varPsi(\cdot,t)\right)\left(\mathcal{F}_{h}\Phi\right),\textbf{g}(\cdot,t) ~\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n},
\end{eqnarray*}
one can say the \textit{discrete convolution operation} (\ref{discreteConvolution}) is well-defined.

\section{Fractional semidiscrete Dirac operators of L\'evy-Leblond type}\label{SemidiscreteSection}
\subsection{The time-fractional case}\label{TimeFractionalSection}

Let us now recall the basic setup and results from a series of papers \cite{FaustinoKGordonDirac16,FaustinoMMAS17,F19,F19b} and also discuss further aspects of our
construction. In the sequel we will use the nilpotents ${\mathfrak f}$ and ${\mathfrak f}^\dagger$ of $C \kern -0.1em \ell_{1,1}$
to introduce firstly on ${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}$, with $C \kern -0.1em \ell_{n+1,n+1}=C \kern -0.1em \ell_{1,1}\otimes C \kern -0.1em \ell_{n,n}$, the semidiscrete Dirac type operator
\begin{eqnarray}
	\label{SemidiscreteDh}
	{~}_\theta\mathcal{D}_{h,t}:=D_h+{\mathfrak f} \partial_t +{\mathfrak f}^\dagger e^{-i\theta},
\end{eqnarray}
carrying the discrete Dirac operator
\begin{eqnarray}
	\label{DiracEqh}
	\begin{array}{lll}
		D_h \Psi(x,t)&=&\displaystyle \sum_{j=1}^n{\bf e}_j\frac{\Psi(x+h {\bf e}_j,t)-\Psi(x-h{\bf e}_j,t)}{2h}+ \\
		&+&	\displaystyle \sum_{j=1}^n{\bf e}_{n+j}\frac{2\Psi(x,t)-\Psi(x+h {\bf e}_j,t)-\Psi(x-h {\bf e}_j,t)}{2h},
	\end{array}
\end{eqnarray}
the time derivative $\partial_t$ and the unitary group parameter $\theta \mapsto e^{-i\theta}$.

From the factorization property $(D_h)^2=-\Delta_h$ involving the \textit{discrete Laplacian} (cf.~\cite[Proposition 2.1]{FaustinoKGordonDirac16})
\begin{eqnarray}
	\label{discreteLaplacian}
	\displaystyle \Delta_h \varPsi(x,t)=\sum_{j=1}^n
	\frac{\varPsi(x+h{\bf e}_j,t)+\varPsi(x-h{\bf e}_j,t)-2\varPsi(x,t)}{h^2},
\end{eqnarray}
and from the graded anti-commuting relations (\ref{CliffordWittBasis}), we can easily prove that
\begin{eqnarray*}
\left({~}_\theta\mathcal{D}_{h,t}\right)^2= e^{-i\theta}\partial_t-\Delta_h.
\end{eqnarray*}

We notice that the square $\left({~}_\theta\mathcal{D}_{h,t}\right)^2$ of (\ref{SemidiscreteDh}) is, up to the unitary term $e^{-i\theta}$, the semidiscrete heat operator considered in \cite{BBRS14}.

Accordingly to \cite{FV16,FV17,BRS20}, a time-fractional counterpart of (\ref{SemidiscreteDh}) can be easily obtained by replacing the time-derivative by a fractional analogue. In particular, based on the Riemann-Liouville derivative (\ref{RiemannLiouville}) we introduce the time-fractional regularization of ${~}_\theta\mathcal{D}_{h,t}$, defined for $\beta\geq 1$ by the {\it time-fractional semidiscrete} operator
\begin{eqnarray}
	\label{TimeFractionalDh}
	{~}_\theta\mathbb{D}_{h,t}^\beta:=D_h-{\mathfrak f} e^{i\theta(1-\beta)}\mathbb{D}_t^\beta +{\mathfrak f}^\dagger e^{-i\theta}
\end{eqnarray}

Thereby, one can easily obtain the factorization property
\begin{eqnarray*}
	\left({~}_\theta\mathbb{D}_{h,t}^\beta\right)^2=-e^{-i\theta\beta}\mathbb{D}_t^\beta-\Delta_h
\end{eqnarray*}
for (\ref{TimeFractionalDh}) in terms of the {\it time-fractional semidiscrete} operator $-e^{-i\theta\beta}\mathbb{D}_t^\beta-\Delta_h$.

\subsection{Time-fractional case vs. space-fractional case}\label{SpaceFractionalSection}

By taking into account the \textit{discrete Fourier transform} (\ref{discreteFh}), one can obtain an equivalent formulation of  ${~}_\theta\mathbb{D}_{h,t}^\beta$ resp.  $-e^{-i\theta\beta}\partial_t-\Delta_h$ in terms of the Fourier multipliers (cf.~\cite[Section 3.3]{F19b})
\begin{eqnarray}
	\label{FourierMultipliers}
	\begin{array}{lll}
		d_h(\xi)^2&=&\displaystyle \sum_{j=1}^{n}\frac{4}{h^2}\sin^2\left(\frac{h\xi_j}{2}\right) \\ \ \\
		\textbf{z}_{h}(\xi)
		&=&\displaystyle \sum_{j=1}^n -i{\bf e}_j\dfrac{\sin(h\xi_j)}{h}+\sum_{j=1}^n {\bf e}_{n+j}\dfrac{1-\cos(h\xi_j)}{h}
	\end{array}
\end{eqnarray}
of $\mathcal{F}_h\circ (-\Delta_h)\circ \mathcal{F}_h^{-1}$ and $\mathcal{F}_h\circ D_h\circ \mathcal{F}_h^{-1}$. Here, we emphasize that the Fourier multiplier formulation lead to the following mapping properties (cf.~\cite[subsection 21.2.2]{F19})
\begin{eqnarray*}
D_h:\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})\rightarrow \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}) \\
-\Delta_h:\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})\rightarrow \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})
\end{eqnarray*}
in a way that
\begin{eqnarray*}
\mathcal{F}_h\left[\left(-e^{-i\theta\beta}\mathbb{D}_t^\beta-\Delta_h\right)\Psi(x,t)\right]&=&\left(-e^{-i\theta\beta}\mathbb{D}_t^\beta+d_h(\xi)^2\right)\mathcal{F}_h\Psi(\xi,t),
	\\
\mathcal{F}_h\left[{~}_\theta\mathbb{D}_{h,t}^\beta \Psi(x,t)\right]&=&	\left(\textbf{z}_{h}(\xi)-{\mathfrak f} e^{i\theta(1-\beta)}\mathbb{D}_t^\beta+{\mathfrak f}^\dagger e^{-i\theta}\right)\mathcal{F}_h\Psi(\xi,t).
\end{eqnarray*}

Thereby, by mimecking \cite[Theorem 4.1 \& Corollary 4.2]{CSV07} the following theorem yields straightforwardly at the level of the Schwarz class $\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$.
\begin{theorem}\label{CoupledSystemParabolicDiracRLt}
	For every $t\in [0,\infty)$ let us assume that let us assume that the components $\Psi^{[m]}(x,t)$ ($m=0,1,2,3$) of the
	${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}-$valued function \begin{eqnarray*}
		\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak f}\Psi^{[1]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[3]}(x,t)
	\end{eqnarray*}
	satisfy the set of conditions ($m=0,1,2,3$)
	\begin{eqnarray*}
		\Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}) &\&& \mathbb{D}_t^\beta \Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}).
	\end{eqnarray*}

Then, for every $\beta\geq 1$ the function $\Psi(x,t)$ is a null solution of ${~}_\theta\mathbb{D}_{h,t}^{\beta}$ (see eq.~(\ref{TimeFractionalDh})) if, and only if
	\begin{eqnarray}
		\label{CoupledSystemsHeatAlphaRL} \left\{\begin{array}{lll}
			{\mathbb D}_t^\beta\Psi^{[m]}(x,t)=-e^{i\theta\beta}\Delta_h\Psi^{[m]}(x,t) & \mbox{for} & m=0,2
			\\ \ \\
			\Psi^{[1]}(x,t)=-e^{i\theta}D_h\Psi^{[0]}(x,t) & &\\ \ \\
			\Psi^{[3]}(x,t)=e^{i\theta}D_h\Psi^{[2]}(x,t)-\Psi^{[0]}(x,t). &  &
		\end{array}\right.
	\end{eqnarray}
\end{theorem}

\begin{remark}
Under the conditions of {\bf Theorem \ref{CoupledSystemParabolicDiracRLt}} one can say that the ${\mathbb C}\otimesC \kern -0.1em \ell_{n,n}-$valued components $\Psi^{[0]}(x,t)$ and $\Psi^{[2]}(x,t)$ of $\Psi(x,t)$ are null solutions of the {\it time-fractional semidiscrete operator} $-e^{-i\theta\beta}{\mathbb D}_t^\beta-\Delta_h$.
\end{remark}

In the space-fractional case one has to consider the \textit{discrete Fourier transform} $\mathcal{F}_h$ and its inverse $\mathcal{F}_h^{-1}$, defined viz eq.~(\ref{FourierInversion}), to introduce a space-fractional regularization of the semidiscrete Dirac ${~}_\theta\mathcal{D}_{h,t}$. Namely, with the aid of the fractional discrete operator $(-\Delta_h)^{\sigma}=\mathcal{F}_h^{-1}\circ \left(d_h(\xi)^2\right)^\sigma \circ \mathcal{F}_h$:
\begin{eqnarray}
	\label{fractionalDeltah}(-\Delta_h)^{\sigma}\Psi(x,t)&=&\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n} \left(d_h(\xi)^2\right)^\sigma \mathcal{F}_h\Psi(\xi,t)e^{-ix\cdot \xi}d\xi,
\end{eqnarray}
 one can define
\begin{eqnarray}
	\label{SpaceFractionalDh}{~}_\theta\mathcal{D}_{h,t}^\alpha :=D_h+ {\mathfrak f}\partial_t(-\Delta_h)^{1-\alpha} +e^{-i\theta} {\mathfrak f}^\dagger
\end{eqnarray}
as the space-fractional regularization underlying to (\ref{SemidiscreteDh}).

We note already that the factorization property
$$({~}_\theta\mathcal{D}_{h,t}^\alpha)^2=e^{-i\theta}\partial_t(-\Delta_h)^{1-\alpha}-\Delta_h$$
is rather immediate from the set of identities
\begin{eqnarray*}
	\left(\textbf{z}_{h}(\xi)+ (d_h(\xi)^2)^{1-\alpha}{\mathfrak f} \partial_t +e^{-i\theta\alpha}{\mathfrak f}^\dagger\right)^2=(d_h(\xi)^2)^{1-\alpha}e^{-i\theta\alpha}\partial_t+d_h(\xi)^2
\end{eqnarray*}
involving the Fourier multipliers of
\begin{eqnarray*}
	\mathcal{F}_h\circ {~}_\theta\mathcal{D}_{h,t}^\alpha \circ \mathcal{F}_h^{-1} & \& & \mathcal{F}_h\circ\left(e^{-i\theta}\partial_t(-\Delta_h)^{1-\alpha}-\Delta_h\right) \circ \mathcal{F}_h^{-1}.
\end{eqnarray*}

Regarding the definition of $(-\Delta_h)^\sigma$ we would like to stress the tangible connection between eq.~(\ref{fractionalDeltah}) and the Bochner's definition (cf.~\cite[Section 6]{CiaurriGRTV17}). Indeed, from the fact that the Fourier symbol $e^{-td_h(\xi)^2}$ belongs to $C^\infty\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}\right)$ one can moreover show that $(-\Delta_h)^{\sigma}$ admits, for every of $0<\sigma<1$, a {\it semidiscrete heat semigroup representation} in terms of $\{e^{s\Delta_h}\}_{s\geq 0}$. Namely, a wise adaptation of the proof of \cite[Lemma 6.5.]{LizamaRoncal18} (see also \cite[Section 2.]{CiaurriGRTV18}):
\begin{eqnarray*}
	(d_h(\xi)^2)^\sigma= \int_{0}^{\infty} g_{-\sigma}(s)\left(e^{-sd_h(\xi)^2}-1\right)ds, &  g_{-\sigma}(s)=\dfrac{s^{-1-\sigma}}{\Gamma(-\sigma)}~  [\mbox{see eq.}~(\ref{GelfandShilov})]
\end{eqnarray*}
allows us to guarantee that $(d_h(\xi)^2)^\sigma$ also belongs to $C^\infty\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}\right)$
and moreover that, $(-\Delta_h)^\sigma$ represented as follows:
	\begin{eqnarray*} (-\Delta_h)^\sigma =\left\{\begin{array}{lll}
		\displaystyle \int_{0}^{\infty} g_{-\sigma}(s)\left(e^{s\Delta_h}-I\right)ds&,~~0<\sigma<1
		\\ \ \\
		-\Delta_h &,~~\sigma=1
	\end{array}\right.
\end{eqnarray*}
satisfies the mapping property
$$(-\Delta_h)^{\sigma}:\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})\rightarrow \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}).$$

Thus, the space-fractional regularization ${~}_\theta\mathcal{D}_{h,t}^\alpha$ of ${~}_\theta\mathcal{D}_{h,t}$ defined viz eq. (\ref{SpaceFractionalDh}) is well defined.
The foregoing relation permits us to derive the following integro-differential-difference representations involving the {\it space-fractional semidiscrete} operators ${~}_\theta\mathcal{D}_{h,t}^\alpha$ and $e^{-i\theta}\partial_t(-\Delta_h)^{1-\alpha}-\Delta_h$, respectively:
\begin{lemma}
			For every $t\in [0,\infty)$ let us assume that the components $\Psi^{[m]}(x,t)$ ($m=0,1,2,3$) of the
			${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}-$valued function  \begin{eqnarray*}
		\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak f}\Psi^{[1]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[3]}(x,t)
	\end{eqnarray*}
	satisfy the set of conditions
	\begin{eqnarray*}
		\Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}) &\&& \partial_t\Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}).
	\end{eqnarray*}

	Then, for every $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$, the componentwise action
	$${~}_\theta\mathcal{D}_{h,t}^\alpha\Psi(x,t)=D_h\Psi(x,t)+ {\mathfrak f}\partial_t (-\Delta_h)^{1-\alpha}\Psi(x,t)+{\mathfrak f}^\dagger e^{-i\theta}\Psi(x,t)$$ admits the integro-differential-difference representation
	\begin{eqnarray*}
		\label{SpaceFractionalDirac} \left\{\begin{array}{lll}
			D_h\Psi(x,t)+\displaystyle \int_{0}^{\infty} g_{\alpha-1}(s)\left(e^{s\Delta_h}-I\right){\mathfrak f}\partial_t\Psi(x,t)ds+{\mathfrak f}^\dagger e^{-i\theta}\Psi(x,t),~~0<\alpha<1 &
			\\ \ \\
			D_h\Psi(x,t)+{\mathfrak f}\partial_t\Psi(x,t)+{\mathfrak f}^\dagger e^{-i\theta}\partial_t\Psi(x,t) ,~~\alpha=1
		\end{array}\right.
	\end{eqnarray*}
	whereas
	\begin{eqnarray*}
		\label{SpaceFractionalLaplacian} \left\{\begin{array}{lll}
			\displaystyle \int_{0}^{\infty} g_{\alpha-1}(s)\left(e^{s\Delta_h}-I\right)e^{-i\theta}\partial_t\Psi(x,t)ds-\Delta_h\Psi(x,t)&,~~0<\alpha<1
			\\ \ \\
		 e^{-i\theta}\partial_t\Psi(x,t)-\Delta_h\Psi(x,t) &,~~\alpha=1
		\end{array}\right.
	\end{eqnarray*}
	stands for the integro-differential-difference representation of $$e^{-i\theta}\partial_t(-\Delta_h)^{1-\alpha}\Psi(x,t)-\Delta_h\Psi(x,t)={~}_\theta\mathcal{D}_{h,t}^\alpha\left({~}_\theta\mathcal{D}_{h,t}^\alpha\Psi(x,t)\right).$$
\end{lemma}

Similarly to {\bf Theorem \ref{CoupledSystemParabolicDiracRLt}} one can also obtain the following the characterization for the {\it space-fractional semidiscrete operator}
${~}_\theta\mathcal{D}_{h,t}^\alpha$ defined viz eq.~(\ref{SpaceFractionalDh}). Although the definition of ${~}_\theta\mathcal{D}_{h,t}^\alpha$ relies essentially on the replacement $$-e^{i\pi \theta(1-\beta)}~{\mathbb D}_t^\beta\longrightarrow \partial_t(-\Delta_h)^{1-\alpha}$$
on the right-hand side of (\ref{TimeFractionalDh}), its proof involves a lot of 	technicalities far beyond the Schwarz class $\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$.
\begin{theorem}\label{CoupledSystemParabolicDiract}
		For every $t\in [0,\infty)$ let us assume that the components $\Psi^{[m]}(x,t)$ ($m=0,1,2,3$) of the
		${\mathbb C}\otimesC \kern -0.1em \ell_{n+1,n+1}-$valued function \begin{eqnarray*}
		\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak f}\Psi^{[1]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[3]}(x,t),
	\end{eqnarray*}
	satisfy the set of conditions
	\begin{eqnarray*}
		\Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}) &\&& \partial_t\Psi^{[m]}(\cdot,t)\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n}).
	\end{eqnarray*}

	Then, for every $0<\alpha\leq 1$ and $|\theta|\leq\frac{\alpha\pi}{2}$, the function $\Psi(x,t)$ is a null solution of ${~}_\theta\mathcal{D}_{h,t}^{\alpha}$ (see eq.~(\ref{SpaceFractionalDh})) if, and only if
	\begin{eqnarray}
		\label{CoupledSystemsHeatAlpha} \left\{\begin{array}{lll}
			\partial_t\Psi^{[m]}(x,t)=-e^{i\theta}(-\Delta_h)^\alpha\Psi^{[m]}(x,t) & , & m=0,2
			\\ \ \\
			\Psi^{[1]}(x,t)=-e^{i\theta}D_h
			\Psi^{[0]}(x,t) & &\\ \ \\
			\Psi^{[3]}(x,t)=e^{i\theta}D_h\Psi^{[2]}(x,t)-\Psi^{[0]}(x,t). &  &
		\end{array}\right.
	\end{eqnarray}
\end{theorem}

\proof
First, we recall that from (\ref{DiracEqh}) and (\ref{WittBasis}), it is rather immediate that the discrete Dirac operator $D_h$ and the Witt basis ${\mathfrak f},{\mathfrak f}^\dagger$ of $C \kern -0.1em \ell_{1,1}$ satisfy the set of relations
\begin{eqnarray*}
	{\mathfrak f} ~D_h=-D_h~{\mathfrak f}, & {\mathfrak f}^\dagger ~D_h=-D_h~{\mathfrak f}^\dagger &~~\mbox{\&}~~ ~D_h~{\mathfrak f}\f^\dagger={\mathfrak f}\f^\dagger~ D_h.
\end{eqnarray*}

Then, by letting act the operator (\ref{SpaceFractionalDh}) on $\Psi(x,t)$, one gets by a straightforwardly computation based on the aforementioned relations that
\begin{eqnarray*}
{~}_\theta \mathcal{D}_{h,t}^\alpha\Psi(x,t)&=&	\left(D_h+{\mathfrak f} \partial_t(-\Delta_h)^{1-\alpha}+ {\mathfrak f}^\dagger~e^{-i\theta}\right)\Psi(x,t) \\
	&=&D_h\Psi^{[0]}(x,t)-{\mathfrak f} D_h\Psi^{[1]}(x,t)\\
	&-&{\mathfrak f}^\dagger D_h\Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger D_h\Psi^{[3]}(x,t)
	\\
	&+&{\mathfrak f}\partial_t(-\Delta_h)^{1-\alpha}\Psi^{[0]}(x,t)\\
	&+&{\mathfrak f}\f^\dagger\partial_t(-\Delta_h)^{1-\alpha}\Psi^{[2]}(x,t)\\
	&+&{\mathfrak f}^\dagger\left(e^{-i\theta}\Psi^{[0]}(x,t)+e^{-i\theta}\Psi^{[3]}(x,t)\right)\\
	&+&(1-{\mathfrak f}\f^\dagger)e^{-i\theta}\Psi^{[1]}(x,t).
\end{eqnarray*}

By rearranging the above identity, one gets the above ansatz written as a linear combination in terms of $1,{\mathfrak f},{\mathfrak f}^\dagger$ and ${\mathfrak f}\f^\dagger$. Namely, one has
\begin{eqnarray*}
{~}_\theta \mathcal{D}_{h,t}^\alpha\Psi(x,t) = \left(D_h\Psi^{[0]}(x,t)+e^{-i\theta}\Psi^{[1]}(x,t)\right) \\
	+{\mathfrak f}\left(\partial_t(-\Delta_h)^{1-\alpha}\Psi^{[0]}(x,t)-D_h\Psi^{[1]}(x,t)\right)\\
	+{\mathfrak f}^\dagger\left(-D_h\Psi^{[2]}(x,t)+e^{-i\theta}\Psi^{[0]}(x,t)+e^{-i\theta}\Psi^{[3]}(x,t)\right)
	\\
	+{\mathfrak f}\f^\dagger\left(\partial_t(-\Delta_h)^{1-\alpha}\Psi^{[2]}(x,t)+D_h\Psi^{[3]}(x,t)-e^{-i\theta}\Psi^{[1]}(x,t)\right).
\end{eqnarray*}

Thus, simply the observation that $$(-\Delta_h)^{\alpha-1}=\mathcal{F}_h^{-1}\circ (d_h(\xi)^2)^{\alpha-1}\circ \mathcal{F}_h$$ is the inverse of $(-\Delta_h)^{1-\alpha}=\mathcal{F}_h^{-1}\circ (d_h(\xi)^2)^{1-\alpha}\circ \mathcal{F}_h$ that yields from definition (\ref{fractionalDeltah}), allows us to conclude that $\Psi(x,t)$ solves equation ${~}_\theta \mathcal{D}_{h,t}^\alpha\Psi(x,t)=0$ if, and only if
\begin{eqnarray}
	\label{CoupledSystemsDiracAlpha} \left\{\begin{array}{lll}
		\Psi^{[1]}(x,t)=-e^{i\theta}D_h\Psi^{[0]}(x,t) & (\mbox{eq.}~1) &
		\\ \ \\
		\partial_t\Psi^{[0]}(x,t)=(-\Delta_h)^{\alpha-1}D_h\Psi^{[1]}(x,t) & (\mbox{eq.}~{\mathfrak f})&\\ \ \\
		\Psi^{[3]}(x,t)=e^{i\theta}D_h\Psi^{[2]}(x,t)-\Psi^{[0]}(x,t) &   (\mbox{eq.}~{\mathfrak f}^\dagger) & \\ \ \\
		\partial_t\Psi^{[2]}(x,t)=-(-\Delta_h)^{\alpha-1}D_h\Psi^{[3]}(x,t)+e^{-i\theta}(-\Delta_h)^{\alpha-1}\Psi^{[1]}(x,t). &   (\mbox{eq.}~{\mathfrak f}\f^\dagger) &
	\end{array}\right.
\end{eqnarray}

Next, using the fact that the discrete Dirac operator $D_h$ (see eq. (\ref{DiracEqh})) satisfies the square property
$(D_h)^2=-\Delta_h$ (cf.~\cite[Proposition 2.1]{FaustinoKGordonDirac16})
we get
\begin{eqnarray*}
	\partial_t\Psi^{[0]}(x,t)&=&-e^{i\theta}(-\Delta_h)^{\alpha-1}(D_h)^2\Psi^{[0]}(x,t) \\
	&=&-e^{i\theta}(-\Delta_h)^\alpha\Psi^{[0]}(x,t)
\end{eqnarray*}
after susbtituting
$(\mbox{eq.}~1)$ on the right-hand side of $(\mbox{eq.}~{\mathfrak f})$.

In the same order of ideas, by substituting
and $(\mbox{eq.}~1)$ \& $(\mbox{eq.}~{\mathfrak f}^\dagger)$ on the right-hand side of $(\mbox{eq.}~{\mathfrak f}\f^\dagger)$, we end up with
\begin{eqnarray*}
	\partial_t\Psi^{[2]}(x,t)&=&-(-\Delta_h)^{\alpha-1}D_h\left(e^{i\theta}D_h\Psi^{[2]}(x,t)-\Psi^{[0]}(x,t)\right)\\
	&+&e^{-i\theta}(-\Delta_h)^{\alpha-1}\left(-e^{i\theta}D_h\Psi^{[0]}(x,t)\right) \\
	&=&-e^{i\theta}(-\Delta_h)^{\alpha-1}(D_h)^2\Psi^{[2]}(x,t) \\
	&=&-e^{i\theta}(-\Delta_h)^{\alpha}\Psi^{[2]}(x,t).
\end{eqnarray*}

Thus, we have shown that the coupled system of equations (\ref{CoupledSystemsDiracAlpha}) is equivalent to (\ref{CoupledSystemsHeatAlpha}).
\qed

\begin{remark}
	Under the conditions of {\bf Theorem \ref{CoupledSystemParabolicDiract}} one can say that the ${\mathbb C}\otimesC \kern -0.1em \ell_{n,n}-$valued components $\Psi^{[0]}(x,t)$ and $\Psi^{[2]}(x,t)$ of $\Psi(x,t)$ are null solutions of the {\it space-fractional semidiscrete operator} $e^{-i\theta}\partial_t+(-\Delta_h)^\alpha$.
\end{remark}

\section{Main Results}\label{MainResultsSection}

\subsection{The analytic semigroup $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$}
In this section we will show that the action of the [fractional semidiscrete] analytic semigroup $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0},$ carrying the parameters $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$, allows us to establish a correspondence between the solution representation of two seemingly distinct Cauchy problems, involving the space-fractional and time-fractional semidiscrete Dirac operators studied previously in Section \ref{SemidiscreteSection}.

Before we proceed, it is important to recall some facts regarding the formulation of $\{\exp(-te^{i\theta}(-\Delta_h)^\alpha)\}_{t\geq 0}$ as an analytic semigroup encoded by the product of {\it modified Bessel functions of the first kind} along the same lines of \cite[Subsection 21.4.2]{F19}.
First, notice that in case of $\alpha=1$, $\left\{\exp\left(te^{i\theta}\Delta_h\right)\right\}_{t\geq 0}$ corresponds to an analytic extension of the {\it semidiscrete heat semigroup}, already treated in \cite[Section 6]{LizamaRoncal18} (see {\bf Remark 13}). In concrete, from the \textit{discrete
	convolution formula} (\ref{discreteConvolution})
there holds
\begin{eqnarray}
	\label{discreteConvolutionHeat}\exp\left(te^{i\theta}\Delta_h\right)\Phi(x)&=&\sum_{y \in h{\mathbb Z}^n}  h^n\Phi(y)K(x-y,te^{i\theta}),
\end{eqnarray}
with
\begin{eqnarray}
	\label{IntegralDiscreteHeat}K(x,te^{i\theta})=\frac{1}{(2\pi)^{\frac{n}{2}}} \int_{Q_h} e^{-te^{i\theta}d_h(\xi)^2} e^{-i x \cdot \xi} d\xi.
\end{eqnarray}
Moreover, the closed formula (cf.~\cite[p. 458, eq.~(21.35)]{F19})
\begin{eqnarray*}
	\label{ClosedFormulaBessel} K(x,te^{i\theta})&=&\frac{(2\pi)^{\frac{n}{2}}}{h^{n}}e^{-\frac{2nte^{i\theta}}{h^2}}  I_{\frac{x_1}{h}}\left(\frac{2te^{i\theta}}{h^2}\right)I_{\frac{x_2}{h}}\left(\frac{2te^{i\theta}}{h^2}\right) \ldots I_{\frac{x_n}{h}}\left(\frac{2te^{i\theta}}{h^2}\right),
\end{eqnarray*}
involving the product of {\it modified Bessel functions of the first kind} (cf.~\cite[p. 456, 2.5.40 (3)]{PBM86})
\begin{eqnarray}
\label{ModifiedBessel}I_k(z)=\frac{1}{\pi}\int_{0}^\pi e^{z\cos(\omega)}e^{-ik\omega}d\omega, & |\mbox{arg}(z)|<\pi.
\end{eqnarray}
results from the identity associated to the Fourier multipliers $d_h(\xi)^2$ (see eq.~(\ref{FourierMultipliers})):
$$
d_h(\xi)^2=\sum_{j=1}^n \frac{2}{h^2}\left(1-\cos(h\xi_j)\right),
$$
and from the change of variables $\xi_j=\frac{\omega_j}{h}$ ($-\pi <\omega_j\leq \pi$) on (\ref{IntegralDiscreteHeat}).

	Because of $|\mbox{arg}(z)|=|\theta|$ for $z=te^{i\theta}$ one can moreover say that $K(x,te^{i\theta})$, described as above, is is well defined for every $|\theta|\leq \frac{\alpha \pi}{2}$ ($\alpha=1$). This conclusion is rather immediate from the integral representation~(\ref{ModifiedBessel}).

We can moreover extend the above analysis to $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$ in case of $0<\alpha<1$ and $|\theta|\leq \frac{\alpha \pi}{2}$. At this stage, it is interesting to see that the Laplace identity
\begin{eqnarray}
\label{Levy}e^{-s^\alpha}=\int_{0}^\infty e^{-rs}L_\alpha^{-\alpha}(r)dr, & \Re(s)>0, & 0<\alpha<1
\end{eqnarray}
involving the {\it L\'evy stable distribution} $L_\alpha^{-\alpha}(r)$ (cf.~\cite[Section 4]{Mainardi01}) provides us to establish a bridge between
\begin{center}
$\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$ and $\left\{\exp\left(te^{\frac{i\theta}{\alpha}}\Delta_h\right)\right\}_{t\geq 0}$,
\end{center}
respectively.

Indeed, for $s=te^{i\theta}(d_h(\xi)^2)^\alpha$($t>0$ is assumed to fix the constraint $\mbox{Re}(s)>0$) one can see from the change of variable $r\rightarrow rt^{-\frac{1}{\alpha}}$ on the right-hand side of (\ref{Levy}) that the underlying Fourier symbols $e^{-te^{i\theta}(d_h(\xi)^2)^\alpha}$ and $e^{-te^{\frac{i\theta}{\alpha}}d_h(\xi)^2}$, respectively, are linked by the integral formula
\begin{eqnarray*}
e^{-te^{i\theta}(d_h(\xi)^2)^\alpha}=\int_{0}^{\infty} e^{-re^{\frac{i\theta}{\alpha}}d_h(\xi)^2}f_{\alpha,\theta}(r) du,~~f_{\alpha,\theta}(r)=t^{-\frac{1}{\alpha}}L_\alpha^{-\alpha}\left(rt^{-\frac{1}{\alpha}}\right).
\end{eqnarray*}

The above representation is useful in several applications on the crossroads of fractional calculus and stochastics, but we will not need to apply it in concrete throughout the present paper. For the interested reader, we refer e.g. to \cite[Chapter 3 \& Chapter 6]{MS12}.

\subsection{Space-fractional vs time-fractional Cauchy problems}

 For the remainder part of this paper we will restrict ourselves to the analysis of $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$ in terms of its symbol $e^{-te^{i\theta}(d_h(\xi)^2)^\alpha}$.
 First of all, we will start to show that the technique used in \cite{CLRV15} to prove {\bf Theorem 3} can be generalized to the fractional semidiscrete analytic semigroup $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$.
\begin{theorem}\label{SemigroupCauchySolver}
	Let $\Phi_0\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$ be given. Then,
	for every $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$, the function
$$
\Phi(x,t)=\exp(-te^{i\theta}(-\Delta_h)^\alpha)\Phi_0(x)
$$
solves the following two Cauchy problems:
\begin{eqnarray}
	\label{CauchySemidiscrete} \left\{\begin{array}{lll}
		\partial_t\Phi(x,t)= -e^{i\theta }(-\Delta_h)^\alpha \Phi(x,t) & \mbox{for} & (x,t)\in
		h{\mathbb Z}^n \times [0,\infty)
		\\ \ \\
		\Phi(x,0)=\Phi_0(x) & \mbox{for} & x\in h{\mathbb Z}^n
	\end{array}\right.
\end{eqnarray}
and
\begin{eqnarray}
	\label{CauchyTimeFractional} \left\{\begin{array}{lll}
		\mathbb{D}_t^{\frac{1}{\alpha}}\Phi(x,t)= -e^{\frac{i\theta}{\alpha}}\Delta_h \Phi(x,t) & \mbox{for} & (x,t)\in
		h{\mathbb Z}^n \times [0,\infty)
		\\ \ \\
		\Phi(x,0)=\Phi_0(x) & \mbox{for} & x\in h{\mathbb Z}^n
	\end{array}\right.
\end{eqnarray}
\end{theorem}

\proof
By letting act the discrete Fourier transform $\mathcal{F}_h$, we obtain an equivalent formulation for the Cauchy problem on the momentum space $Q_h\times [0,\infty)$:
\begin{eqnarray}
	\label{CauchySemidiscreteFourier} \left\{\begin{array}{lll}
		\partial_t\left[\mathcal{F}_h\Phi(\xi,t)\right]= -e^{i\theta}(d_h(\xi)^2)^{\alpha} \mathcal{F}_h\Phi(\xi,t) & \mbox{for} & (\xi,t)\in
		Q_h \times [0,\infty)
		\\ \ \\
		\mathcal{F}_h\Phi(\xi,0)=\mathcal{F}_h\Phi_0(\xi) & \mbox{for} & \xi \in Q_h
	\end{array}\right.
\end{eqnarray}
so that
\begin{eqnarray}
\label{SolutionSemidiscreteFourier}\mathcal{F}_h\Phi(\xi,t)= e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha} \mathcal{F}_h\Phi_0(\xi)& ,(\xi,t)\in Q_h \times [0,\infty)
\end{eqnarray}
solves (\ref{CauchySemidiscreteFourier}), and whence,
\begin{eqnarray*}
\Phi(x,t)&=&\exp(-te^{i\theta}(-\Delta_h)^\alpha)\Phi_0(x)\\
&=&\displaystyle \frac{1}{(2\pi)^{\frac{n}{2}}}\int_{Q_h} e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha} \mathcal{F}_h\Phi_0(\xi) e^{-ix\cdot \xi} d\xi
\end{eqnarray*}
solves (\ref{CauchySemidiscrete}).

Thus, in order to show that $\Phi(x,t)$ is also a solution of the Cauchy problem (\ref{CauchyTimeFractional}), it suffices to show that (\ref{SolutionSemidiscreteFourier}) solves
\begin{eqnarray}
	\label{CauchyTimeFractionalFourier} \left\{\begin{array}{lll}
		\mathbb{D}_t^{\frac{1}{\alpha}}\left[\mathcal{F}_h\Phi(\xi,t)\right]=e^{\frac{i\theta}{\alpha}} d_h(\xi)^2\mathcal{F}_h\Phi(\xi,t) & \mbox{for} & (\xi,t)\in Q_h \times [0,\infty)
		\\ \ \\
		\mathcal{F}_h\Phi(\xi,0)=\mathcal{F}_h\Phi_0(\xi) & \mbox{for} & \xi\in Q_h.
	\end{array}\right.,
\end{eqnarray}
or equivalently, that $$\mathbb{D}_t^{\frac{1}{\alpha}}\left[e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}\right]=e^{\frac{i\theta}{\alpha}}d_h(\xi)^2~e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}$$
holds for every $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$.

To do so, recall that for every $k\in \mathbb{N}$ one has the derivation rule
\begin{eqnarray}
\label{dkExp}\left(-\partial_t\right)^{k}\left[e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}\right]=e^{i \theta  k}\left(d_h(\xi)^2\right)^{\alpha k}e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}.
\end{eqnarray}

In case of $\alpha=\frac{1}{k}$ ($k \in {\mathbb N}$), it readily follows that $\mathbb{D}_t^{\frac{1}{\alpha}}$ equals to $(-\partial_t)^k$, so that
\begin{eqnarray*}
\mathbb{D}_t^{\frac{1}{\alpha}}\left[e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha} \right]&=&(-\partial_t)^k\left[e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha} \right] \\
&=&e^{\frac{i\theta}{\alpha}}
d_h(\xi)^2e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}.
\end{eqnarray*}

Otherwise, set $k=\lfloor \frac{1}{\alpha} \rfloor+1$. Owing the fact that $0<\alpha<1$, the constraint $|\theta|\leq \frac{\alpha \pi}{2}$ assures that the constant $\lambda=e^{i\theta}\left(d_h(\xi)^2\right)^\alpha$ satisfies the condition
$$
\Re(\lambda)=\cos(\theta)\left(d_h(\xi)^2\right)^\alpha>0.
$$

Thus, from the Laplace identity (\ref{LaplaceId}) we get that
$$
\int_0^\infty g_{k-\frac{1}{\alpha}}(p)e^{-pe^{i\theta}\left(d_h(\xi)^2\right)^\alpha}~dp=e^{-i \theta\left(k-\frac{1}{\alpha}\right)}\left(d_h(\xi)^2\right)^{-\alpha\left(k-\frac{1}{\alpha}\right)},
$$
whereby $g_{k-\frac{1}{\alpha}}(p)$ stands for the Gel'fand-Shilov function (see eq.(\ref{GelfandShilov})).

 Then, the eq. (\ref{dkExp}) together with the previous integral formula give rise to the sequence of identities
\begin{eqnarray*}
\mathbb{D}_t^{\frac{1}{\alpha}}\left[e^{-te^{i\theta }\left(d_h(\xi)^2\right)^\alpha}\right]&=&\left(-\partial_t\right)^{k} \int_t^\infty g_{k-\frac{1}{\alpha}}(s-t)e^{-se^{i\theta}\left(d_h(\xi)^2\right)^\alpha}~ds \\
&=&\left(-\partial_t\right)^{k} \int_0^\infty g_{k-\frac{1}{\alpha}}(p)~e^{-(t+p)e^{i\theta}\left(d_h(\xi)^2\right)^\alpha}~dp \\
&=&\left(-\partial_t\right)^{k}\left[e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha}\right] ~\\
&\times &\int_0^\infty g_{k-\frac{1}{\alpha}}(p)~e^{-pe^{i\theta}\left(d_h(\xi)^2\right)^\alpha}~dp\\
&=&e^{i \theta k}\left(d_h(\xi)^2\right)^{\alpha k}e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha} \\
&\times& e^{-i \theta\left(k-\frac{1}{\alpha}\right)}\left(d_h(\xi)^2\right)^{-\alpha(k-\frac{1}{\alpha})} \\
&=&e^{\frac{i \theta}{\alpha}}d_h(\xi)^2~e^{-te^{i\theta}\left(d_h(\xi)^2\right)^\alpha},
\end{eqnarray*}
concluding the proof of (\ref{CauchyTimeFractionalFourier}).
\qed
\begin{remark}
Essentially, we have shown in {\bf Theorem \ref{SemigroupCauchySolver}} that the analytic semigroup $\left\{\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\right\}_{t\geq 0}$, carrying the parameters $0<\alpha\leq 1$ and $|\theta|\leq\frac{\alpha\pi}{2}$, generates simultaneously solutions for Cauchy problems induced by the fractional semidiscrete operators $e^{-i\theta}\partial_t+(-\Delta_h)^\alpha$ and $-e^{-\frac{i\theta}{\alpha}\theta}{\mathbb D}_t^{\frac{1}{\alpha}}-\Delta_h$, respectively.
\end{remark}

\subsection{Cauchy problems of L\'evy-Leblond type}

\begin{figure}\label{MainResultsPicture}
	\begin{tikzpicture}
		\node[mynode] (v1){\textbf{Initial conditions}\\ \ \\	$\Phi^{[0]}_0,\Phi^{[2]}_0\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$ };

		\node[mynode, right=1.0cm of v1](v5) {{\bf Ansatz function} \\ \ \\
			$\Phi_0(x)={\mathfrak f}^\dagger {\mathfrak f} \Phi_0^{[0]}(x)+{\mathfrak f}^\dagger  \Phi_0^{[2]}(x)$	};
		\node[mynode,below=2.0cm of v5](v3) {{\bf Semigroup solver $\Phi(x,t)$} \\ \ \\
			$\Phi(x,t)={\mathfrak f}^\dagger {\mathfrak f} \Psi^{[0]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)$};
		\node[mynode,below =2.0cm of v1](v2) {
			\textbf{Solutions of the Cauchy problems (\ref{CauchySemidiscreteComponentsAlpha}) \& (\ref{CauchySemidiscreteComponentsAlphaRL})} \\ \ \\ $\Psi^{[0]}(x,t)$ and $\Psi^{[2]}(x,t)$.};

		\node[mynode,below = 2.0cm of v2] (v4){{\bf Solutions of the Cauchy Problems of L\'evy-Leblond type (\ref{CauchyFractionalDiracAlpha}) \& (\ref{CauchyFractionalRLt})}\\ \ \\  $\Psi^{[m]}(x,t)~~(m=0,1,2,3)$ of $\Psi(x,t)$.};

		\draw[-latex] (v5.south) -- node[auto,] {$\exp(-te^{i\theta}(-\Delta_h)^\alpha)\Phi_0(x)$} (v3.north);
		\draw[-latex] (v2.east) -- node[auto,] {} (v3.west);

		\draw[-latex] (v2.south) -- node[auto,] {{\bf Theorems \ref{CoupledSystemParabolicDiracRLt} \& \ref{CoupledSystemParabolicDiract}}} (v4.north);
		\draw[-latex] (v3.south) -- node[auto,] {$\Psi(x,t)=\Psi(x,t)-{\mathfrak f} e^{i\theta} D_h\Phi(x,t)$} (v4.east);
		\draw[-latex] (v5.west) -- node[auto,]{} (v1.east);
		\draw[-latex] (v1.south) -- node[auto,]{{\bf Theorem \ref{SemigroupCauchySolver}}} (v2.north);
	\end{tikzpicture}
	\caption{Schematic proof of {\bf Theorem \ref{LevyLeblondSolutions}}.}
\end{figure}

After the meaningful construction obtained previously in {\bf Theorem \ref{SemigroupCauchySolver}}, we have now gathered the main ingredients to prove the main result of this section. On \textsc{figure 2} is neatly summarized the proof of {\bf Theorem \ref{LevyLeblondSolutions}}, whose starting point relies heavily on
{\bf Theorem \ref{CoupledSystemParabolicDiracRLt}} \& {\bf Theorem \ref{CoupledSystemParabolicDiract}}.

\begin{theorem}\label{LevyLeblondSolutions}
 Let $\Phi^{[0]}_0,\Phi^{[2]}_0\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$ be given, and set
\begin{eqnarray*}
\Phi_0(x)&=&{\mathfrak f}^\dagger{\mathfrak f}\Phi^{[0]}_0(x)+{\mathfrak f}^\dagger\Phi^{[2]}_0(x), \\
	\Phi(x,t)&=&\exp(-te^{i\theta}(-\Delta_h)^\alpha)\Phi_0(x).
\end{eqnarray*}

	Then, for every $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$, the function
	\begin{eqnarray}
		\label{HarmonicConjugatesTheta}\Psi(x,t)&=&\Phi(x,t)-{\mathfrak f} e^{i\theta}D_h\Phi(x,t)
	\end{eqnarray}
	solves the following two Cauchy problems of L\'evy-Leblond type: \begin{eqnarray}
		\label{CauchyFractionalDiracAlpha} \left\{\begin{array}{lll}
			\left(D_h+{\mathfrak f} \partial_t(-\Delta_h)^{1-\alpha}+ {\mathfrak f}^\dagger~e^{-i\theta}\right)\Psi(x,t)=0 & \mbox{for} & (x,t)\in
			h{\mathbb Z}^n \times [0,\infty)
			\\ \ \\
			\Psi(x,0)=\Phi_0(x)
			-{\mathfrak f} e^{i\theta}D_h\Phi_0(x)& \mbox{for} & x\in h{\mathbb Z}^n,
		\end{array}\right.
	\end{eqnarray}
	and
	\begin{eqnarray}
		\label{CauchyFractionalRLt} \left\{\begin{array}{lll}
			\left(D_h-{\mathfrak f}~e^{i\theta \left(1-\frac{1}{\alpha}\right)} {\mathbb D}_t^{\frac{1}{\alpha}}+ {\mathfrak f}^\dagger~e^{-i\theta}\right)\Psi(x,t)=0 & \mbox{for} & (x,t)\in
			h{\mathbb Z}^n \times [0,\infty)
			\\ \ \\
			\Psi(x,0)=\Phi_0(x)
			-{\mathfrak f} e^{i\theta}D_h\Phi_0(x)& \mbox{for} & x\in h{\mathbb Z}^n.
		\end{array}\right.
	\end{eqnarray}

\end{theorem}

\proof Under the assumption that $\Phi^{[0]}_0,\Phi^{[2]}_0\in \mathcal{S}(h{\mathbb Z}^n;{\mathbb C}\otimes C \kern -0.1em \ell_{n,n})$, {\bf Theorem \ref{SemigroupCauchySolver}} asserts that
 the components
\begin{eqnarray*}
	\Psi^{[m]}(x,t)=\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\Phi_0^{[m]}(x) & (m=0,2)
\end{eqnarray*}
of
\begin{eqnarray*}
	\Phi(x,t)=\exp\left(-te^{i\theta}(-\Delta_h)^\alpha\right)\Phi_0(x), &\mbox{with} & \Phi_0(x)={\mathfrak f}^\dagger{\mathfrak f}\Phi^{[0]}_0(x)+{\mathfrak f}^\dagger\Phi^{[2]}_0(x)
\end{eqnarray*}
are solutions of the following set of Cauchy problems ($m=0,2$) in case of $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$:
\begin{eqnarray}
	\label{CauchySemidiscreteComponentsAlpha} \left\{\begin{array}{lll}
		\partial_t\Psi^{[m]}(x,t)= -e^{i\theta}(-\Delta_h)^\alpha \Psi^{[m]}(x,t) & \mbox{for} & (x,t)\in
		h{\mathbb Z}^n \times [0,\infty)
		\\ \ \\
		\Psi^{[m]}_0(x,0)=\Phi_0^{[m]}(x) & \mbox{for} & x\in h{\mathbb Z}^n
	\end{array}\right.
\end{eqnarray}
and \begin{eqnarray}
	\label{CauchySemidiscreteComponentsAlphaRL} \left\{\begin{array}{lll}
		\mathbb{D}_t^\beta\Psi^{[m]}(x,t)= -e^{ \frac{i\theta}{\alpha}}\Delta_h \Psi^{[m]}(x,t) & \mbox{for} & (x,t)\in
		h{\mathbb Z}^n \times [0,\infty)
		\\ \ \\
		\Psi^{[m]}_0(x,0)=\Phi_0^{[m]}(x) & \mbox{for} & x\in h{\mathbb Z}^n
	\end{array}\right.
\end{eqnarray}

Then, from {\bf Theorem \ref{CoupledSystemParabolicDiract}} and {\bf Theorem \ref{CoupledSystemParabolicDiracRLt}} one has that the function
\begin{eqnarray*}
	\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak f}\Psi^{[1]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[3]}(x,t),
\end{eqnarray*}
is simultaneously a solution of the Cauchy problems
(\ref{CauchyFractionalDiracAlpha}) and (\ref{CauchyFractionalRLt}), respectively, if and only if the components $\Psi^{[m]}(x,t)$ ($m=1,3$) of $\Psi(x,t)$ are uniquely determined by
\begin{eqnarray*}
	\Psi^{[1]}(x,t)&=&-e^{i\theta}D_h\Psi^{[0]}(x,t), \\ \Psi^{[3]}(x,t)&=&e^{i\theta}D_h\Psi^{[2]}(x,t)-\Psi^{[0]}(x,t).
\end{eqnarray*}

Hence, from
the set of properties
\begin{eqnarray*}
	{\mathfrak f} ~D_h=-D_h~{\mathfrak f}, & {\mathfrak f}^\dagger ~D_h=-D_h~{\mathfrak f}^\dagger &~~\mbox{\&}~~ ~D_h~{\mathfrak f}\f^\dagger={\mathfrak f}\f^\dagger~ D_h
\end{eqnarray*}
that yield from the combination of (\ref{DiracEqh}) and (\ref{WittBasis}), it readily follows that
\begin{eqnarray*}
	\Psi(x,t)&=&\Psi^{[0]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)-{\mathfrak f}\f^\dagger \Psi^{[0]}(x,t)\\
	&+&e^{i\theta}D_h\left[{\mathfrak f}\Psi^{[0]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[2]}(x,t)\right],
\end{eqnarray*}
whereas from the set of the identities
\begin{eqnarray*}
	\Phi(x,t)&=&\Psi^{[0]}(x,t)+{\mathfrak f}^\dagger\Psi^{[2]}(x,t)-{\mathfrak f}\f^\dagger \Psi^{[0]}(x,t) \\
	-{\mathfrak f} e^{i\theta}D_h\Phi(x,t)&=&e^{i\theta}D_h\left[{\mathfrak f}\Psi^{[0]}(x,t)+{\mathfrak f}\f^\dagger\Psi^{[2]}(x,t)\right]
\end{eqnarray*}
we recognize that $\Psi(x,t)$ may be represented through the compact formula (\ref{HarmonicConjugatesTheta}).
\qed

\section{More on space-fractional semidiscrete Dirac operators}\label{SpaceFractionalRemarks}

With the proof of {\bf Theorem \ref{LevyLeblondSolutions}} it was established an intriguing correspondence between the null solutions of time-fractional resp. space-fractional analogues of the semidiscrete Dirac operator, treated on subsection \ref{TimeFractionalSection} \& subsection \ref{SpaceFractionalSection} and the null solutions of the time-fractional resp. space-fractional regularizations of the semidiscrete Dirac operator (\ref{SemidiscreteDh}).

At this stage we would like to emphasize that contrary to the time-fractional regularization (\ref{TimeFractionalDh}), the corresponding space-fractional regularization (\ref{SpaceFractionalDh}) pf (\ref{SemidiscreteDh}) does not factorize the space-fractional semidiscrete operator $e^{-i\theta}\partial_t+(-\Delta_h)^{\alpha}$, as expected, although the factorization of ${~}_\theta\mathbb{D}_{h,t}^{\alpha}$ and ${~}_\theta\mathcal{D}_{h,t}^{\alpha}$ was never required on the proof of main results of this paper.

To circumvent this gap one can introduce the alternative space-fractional semidiscrete variant
\begin{eqnarray}
	\label{SpaceFractionalVariant}{~}_\theta{\bf D}_{h,t}^\alpha:=D_h(-\Delta_h)^{\frac{\alpha-1}{2}}+{\mathfrak f} \partial_t(-\Delta_h)^{\frac{1-\alpha}{2}}+{\mathfrak f}^\dagger e^{-i\theta}(-\Delta_h)^{\frac{\alpha-1}{2}}
\end{eqnarray}
of ${~}_\theta\mathcal{D}_{h,t}=D_h+{\mathfrak f}\partial_t+{\mathfrak f}^\dagger e^{-i\theta}$, involving the space-fractional operator $(-\Delta_h)^{\sigma}$ and its inverse $(-\Delta_h)^{-\sigma}$ ($\sigma=\frac{1-\alpha}{2}$).

Hereby, the discrete fractional operator $D_h(-\Delta_h)^{-\sigma}$ stands for the hypercomplex extension of the fractional Riesz type transform considered in \cite[Section 6]{CiaurriGRTV17} (see also \cite[subsection 21.4.3]{F19}).

To do so, let us now turn our attention to the fractional integral operator $(-\Delta_h)^{-\sigma}$. From the fact that the Fourier multiplier $(d_h(\xi)^2)^{-\sigma}$ of $\mathcal{F}_h\circ (-\Delta_h)^{-\sigma} \circ \mathcal{F}_h^{-1}$ admits, for $0<\sigma<\frac{1}{2}$,
the following Eulerian Laplace integral representation
\begin{eqnarray*}
(d_h(\xi)^2)^{-\sigma}=\int_0^\infty e^{-pd_h(\xi)^2} g_\sigma(p) dp, & g_\sigma(p)=\dfrac{p^{\sigma-1}}{\Gamma(\sigma)} & [\mbox{see eq.}~(\ref{GelfandShilov})]
\end{eqnarray*}
one can guarantee that the mapping property $$(-\Delta_h)^{-\sigma}:\mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})\rightarrow \mathcal{S}(h{\mathbb Z}^n;{\mathbb C} \otimes C \kern -0.1em \ell_{n,n})$$
carrying the inverse of $(-\Delta_h)^{\sigma}$, fulfils for every $0<\sigma<\frac{1}{2}$.

As a consequence, the reformulation of $(-\Delta_h)^{\frac{1-\alpha}{2}}$ and $(-\Delta_h)^{\frac{\alpha-1}{2}}$ in terms of the discrete convolution property (\ref{discreteConvolution}):
\begin{eqnarray*}
	(-\Delta_h)^{\sigma}\Psi(x,t) &=& (\Psi (\cdot,t) \star_{h} \delta_{h,\sigma})(x)\\
	&:=&\sum_{y\in h{\mathbb Z}^n} h^n \delta_{h,\sigma}(x-y)\Psi(y,t),
\end{eqnarray*}
with
$$
\delta_{h,\sigma}(x-y)=
\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n} \left(d_h(\xi)^2\right)^\sigma e^{-i(x-y)\cdot \xi}d\xi
$$
is well defined for every $0<\sigma<\frac{1}{2}$. In case of $\sigma\rightarrow 0^+$ one has that $\delta_{h,\sigma}(x-y)$ converges to the so-called {\it discrete delta function} on $h{\mathbb Z}^n$. That is,
\begin{eqnarray*}
	\lim_{\sigma\rightarrow 0^+}\delta_{h,\sigma}(x-y)= \left\{\begin{array}{lll}
		\dfrac{1}{h^n}, &\mbox{if} & x=y
		\\ \ \\
		0 &\mbox{if} & x\neq y
	\end{array}\right..
\end{eqnarray*}

From the spectral representation $\mathcal{F}_h\circ {~}_\theta{\bf D}_{h,t}^{\alpha}\circ \mathcal{F}_h^{-1}$ in terms of its multiplier
there holds that
the operators
(\ref{SpaceFractionalVariant}) and ${~}_\theta\mathcal{ D}_{h,t}^{\alpha}$ (see eq.~(\ref{SpaceFractionalDh})) are interrelated by the formula
$$
{~}_\theta{\bf D}_{h,t}^\alpha={~}_\theta\mathcal{D}_{h,t}^\alpha(-\Delta_h)^{\frac{\alpha-1}{2}}.
$$

Noteworthy, we obtain the factorization property
$$({~}_\theta{\bf D}_{h,t}^\alpha)^2=e^{-i\theta}\partial_t+(-\Delta_h)^\alpha.$$

The construction, depicted as above, will eventually be used in a forthcoming paper, bearing in mind a possible exploitation of the techniques used e.g. in \cite{CKS05,CFV08} to obtain hypercomplex formulations of boundary value problems as well as possible applications on the crossroads of PDEs and stochastics (see e.g. the models considered e.g.~in \cite{KL12,CarvalhoPlanas15,ZP17}). Thus, the reason for including this extra results in the present paper is mainly for a preliminary motivation for forthcoming papers.

\section*{Acknowledgments}

This research was supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT), references UIDB/04106/2020 and UIDP/04106/2020.

redpajama_set_name: RedPajamaArXiv


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You are here: Home / Archives for ISIS
Sons of bitches — can't live with 'em, can't live without 'em
August 19, 2014 by Clay Bonnyman Evans
One of the more unflattering qualities of American political debate is the tendency to see everything in the world through a lens of U.S. partisan politics.
Thus when the radical Sunni Islamic State — or, if you will, the Islamic State in Iraq and al-Sham, or ISIS — suddenly emerges from the ongoing chaos in Syria and Iraq to pose a seemingly dire threat to the region, too many Americans just want to talk about whose "fault" it is, along strictly partisan lines.
Former U.S. ally Saddam Hussein shakes the hand of former Defense Secretary Donald Rumsfeld, years before Rumsfeld helped prosecute a war to eliminate the "evil" dictator. Associated Press photo.
President Bush and company led the United States into a dubious, idealistic — and disastrous (I need not add "in hindsight," for like millions of others, I objected months before it began and clearly saw where it was likely to lead) — war in Iraq on the basis of lies in 2003, but complaining about that history gets us nowhere. There is plenty to debate about President Obama's role — not to mention the wishes of the Iraqi government — in pulling out of Iraq and his seemingly rudderless foreign policy, but who cares? Sand under the bridge.
International politics is complicated. And almost by definition, hypocritical.
Left-leaning idealists tend to demand that the United States uncouple from unsavory autocrats, dictators and villains like Saddam Hussein, but as Franklin Roosevelt allegedly declared of Nicaraguan dictator Anastasio Somoza Garcia, "He's a son of a bitch, but he's our son of a bitch."
Ngo Dinh Diem, former president of South Vietnam, for whom being an American son of a bitch didn't turn out so hot.
Of course, being America's son of a bitch is a dangerous game. Just ask Saddam or Ngo Dinh Diem.
Meanwhile the John McCains of America are all realpolitik and seemingly no discernment, eager to unleash bombs on every perceived enemy or sign of instability around the globe, it's the details that matter. New-style libertarians like Rand Paul — not as libertarian as he would like everyone to believe — counsel less involvement all around.
But what would those armchair right-wing screamers and morally outraged liberals have us do?
Take Saddam. The post-Ottoman fiction that is "Iraq," with borders practically scratched out in the sand to suit the imperialistic whims of France and Britain in the Sykes-Picot agreement of 1916 and forcing together three wildly disparate ethno-religious populations, always required a "strongman" to hold it together, whether an invented king or a Baathist dictator.
ISIS is even more illustrative. Obama's critics, from McCain to Hillary Clinton, blast him for not choosing sides in the confusing civil war in Syria, most notably for failing to arm the rebels opposed to President Bashar al-Assad, inarguably a dictatorial thug.
Bashar al-Assad – will he soon be America's son of a bitch?
But guess what? Assad, and his father before him, Hafez al-Assad, represent something much closer to Western values than any of the rebels opposing him. Though certainly willing to use religion to his advantage, he's essentially a secularist. ISIS, so violent that al-Qaida keeps it at arm's length, has roots in al-Qaida in Iraq — which did not exist before the U.S. invasion — and the rebels fighting Assad in Syria.
Now, as ISIS seems to be the most virulent enemy on the scene in Iraq and the Levant (basically, Syria and Lebanon), suddenly Assad looks like a "son of a bitch" we might want to deal with. Ditto for the other bogeyman in the region, Shi'ite Iran, which has an incentive to fight Sunni ISIS. Oh, and the long-time American "son of a bitch" Saudi Arabia bears enormous responsibility for helping to create Sunni radicalism and terrorism.
As Joe Klein wrote recently in TIME magazine, "It will require a strategic rethink of who our friends and enemies are in the region. We may find that Iran is part of the ISIS solution rather than part of the problem–a problem that Saudi Arabia's support for Sunni extremism helped create. We may even find ourselves on the same side as Syria's disgraceful Bashar Assad: ISIS is the greatest threat to his continued rule."
I certainly don't have the answer to such questions, and there probably is no "good" answer. I'm sympathetic to Paul's view that the U.S. needs to disentangle itself from such situations, stop carrying the Western world's military water and recognize where our true interests lie. Yet there is danger in isolationism, too.
But here's something I do know: When it comes to incredibly complex international conflicts that go back decades, centuries, even millennia, finger-pointing over current American partisan politics is an incredibly arrogant, myopic waste of time.
Filed Under: Politics and policy Tagged With: assad, ISIS, joe klein, saddam

redpajama_set_name: RedPajamaCommonCrawl


text:
Tiska Djanet Airport  is an airport serving Djanet, Algeria. The airport is in the desert  south of Djanet.  There is a VOR/DME and an NDB on the field for navigation.

The original Djanet Airport  is  south of Djanet, and is being repurposed as a storage yard.

Airlines and destinations

References

External links
 OurAirports - Djanet Inedbirene

 Etablissement de Gestion de Services Aéroportuaires d'Alger (EGSA Alger)

Airports in Algeria
Buildings and structures in Illizi Province

redpajama_set_name: RedPajamaWikipedia


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Kanye's New Fling Julia Fox Featured in Pete Davidson Photo Shoot in 2019
OMG! Kim Kardashian and Kanye West's respective new romances are heating up amid their divorce. But it appears that their respective new flames, Pete Davidson and Julia Fox, are unexpectedly connected.
Kim Kardashian and Kanye West's Relationship Timeline
In fact, the 28-year-old Saturday Night Live star and the 31-year-old actress played boyfriend and girlfriend Barbie and Ken for a November 2019 Paper Magazine cover story, which was ironically entitled "Break the Internet."
For the photo shoot, Davidson and Fox posed for a series of sexy snaps. Though there's nothing that suggests there's ever been a romantic connection, it's no surprise that they are quickly making their rounds on social media.
Kim Kardashian and Pete Davidson's Romance: Everything We Know
The internet has people freaking out over the unpredictable connection between Kimye's new flames. "Ok this julia fox / pete davidson paper mag shoot was just brought to my attention and i'm about to go full tin foil hat conspiracy theorist," tweeted Vanity Fair writer Emily Kirkpatrick. Another person wrote, "This pete davidson/julia fox photoshoot was an omen."
This news comes as both Kardashian and West are pursuing new love interests. A source told Us that the Yeezy founder's new romance is "casual," but that hasn't stopped them from spending time together.
On Tuesday, January 4, the duo attended Broadway's Slave Play and subsequently dined at Carbone in New York City, according to an eyewitness. "It looked romantic," the insider said. "They were only close to each other coming out of Carbone and they looked like they wanted to be close to each other. It was 100 percent a date."
Days prior, they rang in the new year together in Miami. A source exclusively told Us that the duo was "having fun" during the outing and that West "wanted to get to know her better."
The Keeping Up With the Kardahians alum on the other hand has been making headlines with Davidson since October.
The pair seemingly confirmed their romance when they were photographed holding hands in Palm Springs in November 2021. And in December, an insider told Us that the Skims founder's relationship with Davidson is "getting serious."
In the King of Staten Island star's Paper Magazine interview, he opened up about his love language: "I treat the person I'm with like a princess," he told the outlet. "I try and go as above and beyond as possible, because that's what you're supposed to do? If you're in a relationship with someone, you're just supposed to make that person feel as special as possible."
Pete Davidson's Dating History: Ariana Grande, Kaia Gerber, Kate Beckinsale and More!
Kardashian and West called it quits in February 2021 after six years of marriage. They share children North, Saint, Chicago and Psalm.
How The Talk Addressed Sharon Osbourne's Exit In First Episode Back (Updating Live)
Chrissy Teigen shows off bright purple hair in amazing beauty transformation
Surprise holiday stimulus payment of $6,300 being sent out in city on December 15 – do you qualify?
Heidi Montag Opens Up About Her Struggle To Conceive A Second Child
This Tongue-in-Cheek Monologue Has Me Excited For Netflix's The Gay Agenda — See the Trailer!
'Bridgerton' Season 2 Reveals How Lady Whistledown Always Gets 'The Good Gossip'


text:
The cheerful sounds of kernels popping, children laughing and waves crashing against the shore have filled the streets of Ocean City for years.
But thanks to a new attraction, the noises downtown have gotten a lot more sinister.
From realistic-looking zombies to torture chamber rooms, Ocean City Screams Haunted House has it all.
The haunt depicts one twisted scene after another, telling the tale of a ship lost at sea — stuck somewhere between the living and the dead. The one-of-a-kind attraction, which is quickly becoming known as one of the scariest in the state, features two floors of scream-worthy surprises and more than 100 pieces of animation.
"It's pretty terrifying," said Ellen Lemberger, Ocean City Screams' project and talent manager. "Many people struggle to make it all the way through."
The haunted house was designed and built by Steelhead Events and Productions, a company who began frightening people more than 16 years ago.
What began back then as a small Halloween fundraiser, she said, has grown exponentially to include the widely popular Field of Screams attraction in Olney, Maryland, Scream City in Washington D.C., and now, Ocean City Screams. The company has also been recognized by A&E Network as the Best Haunt Designer and Builder in the United States.
Lemberger began her career with Steelhead nearly two decades ago in Olney, were she worked the concession stand at Field of Screams. For weeks, she refused to partake in the haunted ride, but eventually gave in, and loved it.
"One thing led to another," she said, smiling.
Lemberger no longer hands out the french fries, but instead, hand selects the actors and actresses for each location. It takes her months to audition and hire the right team. And although her position is now considered to be more of "behind the scenes" work, Lemberger has been known to slip on a mask and lurk in the shadows from time to time.
"I've gotten good at freaking people out," she said.
Michelle Clauser is among the dozens of people hired to work at Ocean City Screams.
When she's on duty, she is one of the ship's commanders, covered from head to toe in dried blood and fleshy gashes. A pair of spooky, white contact lenses complete her look.
"Sometimes I forget, and I almost scream myself when I look in the mirror," she said. "It's funny — I've never been one for scary movies, but this is awesome."
Ocean City Screams is open to the public every evening from 6-11 p.m.
Although many have backed out from fright, some as young as 5-years-old have made it through successfully.
"We'll take any age — there is no deadline on fear," said Carlton Meyer, Ocean City Screams' manager.
Meyer, a longtime friend of the owners of Steelhead, began volunteering with the company three years ago.
When the opportunity to work full-time in Ocean City presented itself, he couldn't resist. Although his position mainly involves hyping up the guests and ensuring everything runs smoothly, like Lembeger, Meyer has been known to hide in the darkness and jump out at an unsuspecting soul.
"This is the best job," he said. "I honestly enjoy what I do. And even though people are usually screaming, they are enjoying themselves too. How many people get to say they make other people happy for a living?"
Barbecue and wine? New bill will make it possible
Major entertainment venue coming to Lewes
On Twitter @hcarroll_1 and Facebook at Facebook.com/byhannahcarroll
On Instagram @thesassy_banana
Ocean City Screams
Where: 14 Worcester St, Ocean City
When: 6-11 p.m. daily
Visit: ocscreams.com


South Africa has made considerable progress in overcoming the legacy of decades of official denial of the widespread practice of torture by state security forces. However incidents of these human rights violations do continue to occur, primarily in the context of ordinary criminal investigations, during the arrest and detention of suspected illegal immigrants or "terrorism" suspects, in the police response to public demonstrations and in prisons. The current government's challenges in combating high crime levels mean they must make it clear to law enforcement officials and the public that there can be no justification for torture and other ill-treatment.


Rates: 135-130 USD per night. Minimum Rental - 1 week.
Seasonal seaside cottage for couple halfway between New York City and Boston. Summertime: bask in the sun on the beach, or curl up with a book in the shade in the garden. Autumn: walk the shores, or sit by the fire. Spring: enjoy the bloom of flowers and the salt air. Every season has its own delights here on the Connecticut shore.
Situated at the back of a cluster of houses at 98 Limewood Avenue, the Ennis Cottage is away from street and traffic noise, but within 200 feet of a private beach. The cottage has been completely remodeled and is tastefully furnished and supervised by the owners. In a minute one can sit with a panoramic view of Long Island Sound, between Haycock Point and Indian Neck, looking out toward Green Island and the Thimble Islands. Fishing and boating are excellent. A small park within one mile offers tennis, basketball, and a children's playground. A marketplace, liquor store, laundry, and two restaurants are within walking distance, and a larger mall and churches are with a few miles. The Outlet stores are 1/2 hour away, two casinos are one hour away, IKEA is a 15 minutes and the ferry to Block Island is a one hour drive. The entire house has been renovated. living room has wall-to-wall carpeting, a new full size futon and a love seat, several chairs, bureau, television, VCR, telephone, cathedral ceiling, wireless internet connection, and stove with gas log. The kitchen is new and well equipped with gas stove, microwave and grilling oven, refrigerator, pots and pans, dishes, glasses, and silverware. The bedroom has a double bed with drawers beneath, closet, built in air conditioner. Wall-to-wall carpeting. Towels and sheets are provided The back garden has a patio, gas grill, chairs, and abundant flowers.
TV, DVD, Fireplace, Sleeper Sofa - Queen. Double Bed, Private Bath.
The cottage is tucked behind our house but within a few hundred feet of the road and beach so it is quiet and private. The back garden is fenced in and has a patio with gas grill, table and chairs and lawn recliners. You can be in the center of New Haven with it's historic sights, galleries and Yale University. The casinos and Outlet shops are a short ride away, and the Thimble Island Ferry can take you on a tour of these quaint islands.
Mid (6/18 - 7/1) ( 9/10- 9/23) $940/week.
Cancellation Policy: We require a 50% deposit with balance due 30 days before your arrival. Once we reach that 30 day point, the rental fee is nonrefundable unless we are able to rebook the property. If we rebook the property we will refund a prorated share of the rental fee for each day we are able to fill less a $25 cancellation fee. There is a $200 security deposit which is returned if the place is left clean and undamaged, and a $100 cleaning fee which you can waive if you want to clean yourselves.
Deposit Required for Holiday Cottage : 50% deposit.


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Personnages
Hinojosa — est un patronyme, porté notamment par :
 José María Hinojosa, (°  - † ), est un poète, éditeur et avocat espagnol appartenant à la Génération de 27.

Toponymie
Hinojosa est un nom de lieu, notamment porté par :
 Hinojosa del Duque, ville d'Espagne, dans la province de Cordoue, communauté autonome d'Andalousie.
 Hinojosa de Jarque, commune d'Espagne, dans la Comarque de Cuencas Mineras, province de Teruel, communauté autonome d'Aragon.
 Hinojosa de San Vicente, commune d'Espagne de la province de Tolède dans la communauté autonome de Castille-La Manche.
 Hinojosas de Calatrava, commune d'Espagne de la province de Ciudad Real dans la communauté autonome de Castille-La Manche.
 Hinojosa del Valle, commune d'Espagne, dans la province de Badajoz, communauté autonome d'Estrémadure.

Homonymie


Bank financing is possible, but the process is long and tedious and the terms are often not so favorable with interest rates as high as 13%. However, recently owners of property in Costa Rica have seen that there are definite advantages to offering "owner financing" on their properties for sale. Poor returns on investments like CDs and savings accounts, coupled with the instability of the stock market, have made the potential to earn 6-10% per year to owners very attractive indeed.
Up to 3 points for origination fees.
1.5% in additional closing costs (based on loan amount).
Here in Costa Rica, the banks will NEVER loan more than 80% of a home's value and appraisals are very conservative. Interest rates vary widely, but the range is typically from 8.5% on the low end on up to 13% on the high end. Often, the lower interest rate is an introductory rate that is for the first 3 years, which adjusts to a higher amount in the 4th year. Origination fees can be as high as 3 points, and closing costs up to an additional 1.5% of the loan amount. Thinking about a 30 year mortgage? Forget about it! In Costa Rica your options are either 10 years or 20 years, with 20 years being the maximum. Time is of the essence to you to close? Well, not with banks in Costa Rica. Fast closings with banks do not exist. You can expect the process to take anywhere from 6 weeks to 3 months on average.
OWNER FINANCING - Since the beginning of 2010, we have sold several homes with owner financing and with negotiated terms that are win/win for both the buyer and seller. The good news about owner financing is that it is negotiable, whereas bank financing is not. When you find the right property and make an offer with an owner financing contingency, you will want to start with the terms that are best for you. Of course, one important factor to keep in mind is that you will not be able to negotiate on the price in the same way when dealing with owner financing because "cash is king". In this modern world, cash really talks, but owner financing may give you the options you need to get the job done.
No upfront points or origination fees.
In this particular deal, the buyer got exactly want he wanted, and his existing retirement investments are paying his mortgage. Talk about a great situation!
Now that these facts are on the table, there are a few other factors that you must think about.
Options: When looking for owner financing you limit your options. Your expectations cannot be the same as if you are a cash buyer because the number of homes available to you with owner financing is limited. Only about 1 in 10 properties offer owner financing. There are always compromises and having the option of owner financing is a big plus to many. If you find a property that meets 60-70% of your wants and needs, plus it offers owner financing, then you have found a great deal!
Negotiations: As with everything in life, there is compromise. In general, owners are very conservative when negotiating on price if they are offering financing. For example, on our most recent owner financing deal, the asking price was $355,000.00. After negotiations, the owner accepted $350,000.00 with favorable financing terms.
At the end of the day, owner financing can be an excellent situation for both the buyer and seller. For the buyer, it affords the opportunity of making the dream of owning a vacation or retirement home in Costa Rica a reality, with more favorable terms than a bank. In some cases it may allow a buyer to use existing IRAs and investments to pay their mortgage payments without penalty. For the seller, it widens the pool of buyers and can give a much better return on investment when compared to traditional instruments like CDs, savings accounts, or the volatile stock market. It is important that both parties keep an open mind and be flexible in working out these deals. If buyers are looking for 3-4% interest rates that are being offered "back home", then forget about it because that is not going to happen in Costa Rica where people are not desperate. Likewise, if sellers are expecting a 12-15% return, well that is unrealistic in most cases. Ultimately, there is always a happy medium and our job is to strike that balance and make the deal work for both the buyers and sellers. Everyone has a dream and with our real estate knowledge and experience here at Pure Life Development, we can help make that dream a reality!


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Competing visions for 'social impact' development at 8th and Race
8th and Race parking lot
On Monday night, the battle lines formed around the last large undeveloped lot in Chinatown. The oceanic parking lot at 8th and Race, which also hosts a little used subway stop on the Broad-Ridge Spur, is a hot commodity after lying fallow for decades.
The forces on one side are the Parkway Corporation, in collaboration with the Philadelphia Chinatown Development Corporation (PCDC) and Presbys Inspired Life. They are advancing a design by renowned architect Cecil Baker.
On the other side, affordable housing developer Pennrose in alliance with a constellation of legal aid groups, EZPark, and local hotel developer United Development.
The Philadelphia Redevelopment Authority (PRA) issued a request for proposals in September, the agency's first seeking "social impact" proposals, which drew only these two development proposals.
The two camps made their case to an overflowing crowd at 1234 Market Street on Monday evening. The quiet buzz of individual translators hummed in the background throughout the proceedings, as the presentations were translated into Mandarin, Cantonese, and Vietnamese.
The fiercest oratory came towards the end, as the Pennrose group faced criticism from a fiery audience member who questioned the group's root in the community.
"My question is why is this property going to anybody else but the Chinatown community," asked Cecilia Moy Yep, a legendary advocate and lifelong neighborhood resident.
Both developers met with PCDC after the Redevelopment Authority, which hosted Monday's meeting, released an RFP for the vast parking lot at the northeastern corner of Chinatown. Though the lot is large, a significant portion of the property cannot be built on because of the Broad-Ridge Spur subway running beneath it and it must accommodate the Chinatown station entrance. [PRA spokeswoman Jamila Davis also reminded PlanPhilly that a sewer line and the commuter rail tunnel further complicate the site's buildout.]
PRA declined to release the details of the proposals to the public (although executive director Gregory Heller said they are very similar in price).
But Parkway—which has a history in Chinatown—received the backing of PCDC, which is the dominant community organization in the neighborhood. In advance of this particular meeting, that support manifested in a petition drive urging residents to support affordable housing, and to "stand up against gentrification and reclaim Chinatown."
Parkway's project includes a mix of affordable senior apartments, market rate condos, and retail space that includes a supermarket. It also features an aeroponic urban farm, a cutting-edge technology that feeds plants with nutrient-enriched mists (thus requiring much less water than hydroponic farms). Nothing like it exists in Philadelphia.
The lead presenter, Parkway president Robert Zuritsky, compared their proposal to The Pearl—a Chinatown condo building they completed in 2008. He noted that half of the equity of that project had been raised from Asian investors and that 99 out of 100 units were purchased by Asian buyers.
(courtesy of Parkway Corp.)
The Parkway team estimated the condos proposed for the 8th and Race site would range in price between $300,000 for a one bedroom to three bedrooms starting at $500,000. The 60 units of affordable housing for seniors would be offered at different percentages of the area median income—20 percent, 50 percent, and 60 percent—with price estimates of between $300 and $900 a month.
PCDC's executive director, John Chin, offered a speech in support of the proposal. He said that the CDC's mission is to lift its constituents out of poverty, implying that the Parkway proposal offered the best means to achieve that goal.
Chin also offered that Chinatown is already 50 percent denser than the city as a whole. "Our people need breathing room," he concluded.
The Parkway team repeatedly noted that their project offered shorter buildings, less density, and more parking at the behest of PCDC. None of the buildings would be taller than six stories.
"It's important to note that we are keeping the density of the overall project down," said Zuritsky. "We didn't want to overpower this neighborhood with a gigantic development and then only have a couple parking spaces. We intentionally designed this with guidance from John to have more parking spaces and less density."
A rendering of Pennrose's proposed development in Chinatown. | WRT/Pennrose Properties (Courtesy of Pennrose Development)
The Pennrose design team offered substantially taller buildings, although many of the offerings within them were broadly similar.
Commonalities included ground floor retail, 55-65 units of affordable housing for seniors, and between 150 and 160 units of market rate rental apartments. The presenters estimated studios would rent at $1,500 a month and two bedrooms would go for $2,200.
Both plans proposed basically the same amount of parking spaces, 180 (Pennrose) and 181 (Parkway).
The big difference: Instead of an aeroponic farm and a supermarket, there would be a 147-room hotel and an office building that would unify most of the city's legal aid community under one roof.
It is this latter building, dubbed the Equal Justice Center, that is the centerpiece of Pennrose's design. Pennrose vice president Timothy Henkel called it "our differentiator."
At 160,000 square feet, and 14 stories tall, it would hold 300 employees of 25 non-profits in one place. The Equal Justice Center would house pillars of the legal aid world like Philadelphia Legal Assistance and Community Legal Services, which currently holds increasingly pricey offices on Chestnut Street around the corner from City Hall.
The proposal is meant to both save money for the legal aid organizations, which aren't known for their fat coffers, and for their clients who will be able to travel to one place for many of their legal needs.
The Equal Justice Center would be the tallest building in Pennrose's project. The affordable housing tower would be nine-stories and the market rate rentals in an 11-stories. The hotel, likely a Comfort Inn, would be eight stories tall.
Despite PCDC's heated petition drive in advance of the hearing, the crowd seemed largely sympathetic to both designs.
At the end of the meeting the longtime Chinatown advocate, Yep, forcefully made an appeal to history for PCDC's right to have a hand in developing Chinatown. She also feared the effects it would have on congestion: "My problem with the plan is that it will bring in too much traffic."
Thoai Nguyen, head of the Southeast Asian advocacy group SEAMACC, said that he leaned the other way.
"The tie breaker for me is the Equal Justice Center," said Nguyen. "In the next four years, and perhaps the next eight, our communities are going to be under attack. The Equal Justice Center to me is a no-brainer [for that reason]. But my very naive question is this: Is it possible that whichever proposal is selected would consider a version of the Equal Justice Center?"
The Redevelopment Authority's Heller, who moderated the whole event, concluded the evening. The PRA board would chose between the two proposals based on their design, financing, and the social impact the respective plans would have on the city. After a developer is selected, several months will be spent hammering out an agreement with PRA, and securing approval by City Council.
*Note: An earlier version of this article misspelled Pennrose and mistakenly stated that Pennrose's plan was proposing fewer parking spaces than Parkway. PlanPhilly regrets the error.
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