The Umbral Refinement Equation
نویسندگان
چکیده
منابع مشابه
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...
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In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This qumbral calculus can be used to provide solutions to linear q-difference e...
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We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the general properties of these transformations, derive explicit examples and discuss them in the case of the Appèl and Sheffer polynomial families. We apply these transformations to non-linear equations, and d...
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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.
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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 ...
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