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# Preprint “Special functions of hypercomplex variable on the lattice based on SU(1,1)”

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 $SO(n)\times \mathfrak{su}(1,1)$– 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 $E=\sum_{j=1}^nx_j \partial_{x_j}$.

Moreover, the interpretation of the one-parameter representation $\mathbb{E}_h(t)=\exp(tE_h^--tE_h^+)$ of the Lie group $SU(1,1)$ as a semigroup $\left(\mathbb{E}_h(t)\right)_{t\geq 0}$ will allows us to describe the polynomial solutions of a homogeneous Cauchy problem on $[0,\infty)\times h\mathbb{Z}^n$ involving the differencial-difference operator $\partial_t+E_h^+-E_h^-$.

arXiv preprint: http://arxiv.org/abs/1304.7191v2

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