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 hZn, which are Clifford-vector-valued, we will introduce new families of special functions of hypercomplex variable possessing su(1,1) symmetries. The Fourier decomposition of the space of Clifford-vector-valued polynomials with respect to the SO(n)×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 of the Euler operator E=∑j=1nxj∂xj. Moreover, the interpretation of the one-parameter representation Eh(t)=exp(tE−h−tE+h) of the Lie group SU(1,1) as a semigroup (Eh(t))t≥0 will allows us to describe the polynomial solutions of an homogeneous Cauchy problem on [0,∞)×hZn involving the differencial-difference operator ∂t+E+h−E−h.

Keywords

• Clifford algebras
• finite difference operators
• Lie algebras