  Based on the representation of a set of canonical operators on the lattice
$h\mathbb{Z}^n$, which are Clifford-vector-valued, we will introduce new
families of special functions of hypercomplex variable possessing
$\mathfrak{su}(1,1)$ symmetries. The Fourier decomposition of the space of
Clifford-vector-valued polynomials with respect to the ${\rm SO}(n)\times
\mathfrak{su}(1,1)$-module gives rise to the construction of new families of
polynomial sequences as eigenfunctions of a coupled system involving
forward/backward discretizations $E_h^{\pm}$ of the Euler operator
$E=\sum\limits_{j=1}^nx_j \partial_{x_j}$. Moreover, the interpretation of the
one-parameter representation $\mathbb{E}_h(t)=\exp(tE_h^--tE_h^+)$ of the Lie
group ${\rm SU}(1,1)$ as a semigroup $\left(\mathbb{E}_h(t)\right)_{t\geq 0}$
will allows us to describe the polynomial solutions of an homogeneous Cauchy
problem on $[0,\infty)\times h{\mathbb Z}^n$ involving the
differencial-difference operator $\partial_t+E_h^+-E_h^-$.
