Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)

Abstract: 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 $\partial_t+E_h^+-E_h^-$.