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 $\mathrm{SO}(n)\times \mathfrak{su}(1,1)$-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 $E=\sum _{j=1}^n{x}_j{\partial }_{x_j}$. Moreover, the interpretation of the one-parameter representation ${\mathbb{E}}_h(t)=\exp (t{E}_h^{-}-t{E}_h^{+})$ of the Lie group $\mathrm{SU}(1,1)$ as a semigroup ${\left({\mathbb{E}}_h(t)\right)}_{t\ge 0}$ 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 ${\partial }_t+E_h^{+}-E_h^{-}$.

References in corpus (3)

• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Fischer Decomposition for Difference Dirac Operators
• (Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries

Cited by in corpus (3)

• Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
• A conformal group approach to the Dirac-Kähler system on the lattice
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective