Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

Abstract

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature (0,n) the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function (EGF) carrying the continuum Dirac operator D=∑j=1nej∂xj together with the Lie-algebraic representation of raising and lowering operators acting on the lattice hZn is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson–Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.