Umbral calculus, difference equations and the discrete Schrödinger equation

Umbral Calculus, Difference Equations and the Discrete Schrödinger Equation

We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space–time. In this approach a quantum system on a lattice has a symmetry algebra iso...

Diierence Equations via the Classical Umbral Calculus

Formal Calculus and Umbral Calculus

We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial “associativity” property of an im...

Algebras and the Umbral Calculus ∗

We apply Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of λ-umbral calculi parameterized by λ in the base ring.

$\psi$-Appell polynomials` solutions of an umbral difference nonhomogeneous equation

One discovers why Morgan Ward solution [1] of ψ-difference calculus nonhomogeneous equation ∆ ψ f = ϕ in the form f (x) = n≥1 B n n ψ ! ϕ (n−1) (x) + ψ ϕ(x) + p(x) recently proposed by the present author (see-below)-extends here now to ψ-Appell polynomials case-almost automatically. The reason for that is just proper framework i.e. that of the ψ-Extended Finite Operator Calculus(EFOC) recently ...