Discrete q-derivatives and symmetries of q-difference equations

@article{Levi2003DiscreteQA,
  title={Discrete q-derivatives and symmetries of q-difference equations},
  author={D. Levi and J. Negro and M. A. Olmo},
  journal={Journal of Physics A},
  year={2003},
  volume={37},
  pages={3459-3473}
}
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 properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to… Expand
16 Citations
Umbral calculus, difference equations and the discrete Schrödinger equation
In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. TheExpand
Kneser's theorem in q-calculus
While difference equations deal with discrete calculus and differential equations with continuous calculus, so-called q-difference equations are considered when studying q-calculus. In this paper, weExpand
Quantum mechanics and umbral calculus
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrodinger equation substituting theExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Small divisor problem for an analytic q-difference equation
Abstract Similar to the problem of linearization, the “small divisor problem” also arises in the discussion of invertible analytic solutions of a class of q -difference equations. In this paper weExpand
Small Divisor Problem in Dynamical Systems and Analytic Solutions of a q-Difference Equation with a Singularity at the Origin
AbstractIn this paper, we consider a q-difference equation $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(y(q^jz))^{t}=G(z)$$in the complex field $${\mathbb C,}$$ where Ct,j(z) and G(z) have a h1Expand
Lie Symmetries for Lattice Equations
Lie symmetries has been introduced by Sophus Lie to study differential equations. It has been one of the most efficient way for obtaining exact analytic solution of differential equations. Here weExpand
Construction of q-discrete two-dimensional Toda lattice equation with self-consistent sources
Abstract The q-discrete two-dimensional Toda lattice equation with self-consistent sources is presented through the source generalization procedure. In addition, the Grammtype determinant solutionsExpand
Rota-Baxter operators on pre-Lie algebras
Abstract Rota-Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In thisExpand
Umbral Calculus, Difference Equations and the
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 thenExpand
...
1
2
...

References

SHOWING 1-10 OF 40 REFERENCES
Umbral calculus, difference equations and the discrete Schrödinger equation
In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. TheExpand
Quantum symmetries of q‐difference equations
A general method is presented to determine the symmetry operators of linear q‐difference equations. It is applied to q‐difference analogs of the Helmoltz, heat, and wave equations in diverseExpand
Symmetries of theq-difference heat equation
The symmetry operators of aq-difference analog of the heat equation in one space dimension are determined. They are seen to generate aq-deformation of the semidirect product of sl(2, ℝ) with theExpand
Discrete derivatives and symmetries of difference equations
We show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable for finding the symmetries of discrete equations. InExpand
Lie symmetries of difference equations
The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention it is paid to the Lie structure of these symmetries, as well as to theirExpand
Canonical commutation relation preserving maps
We study maps preserving the Heisenberg commutation relation ab - ba = 1. We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dualExpand
Lie group formalism for difference equations
The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples areExpand
Twisted conformal algebra so(4, 2)
A new twisted deformation, Uz(so(4, 2)), of the conformal algebra of the (3 + 1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by tenExpand
The Umbral Calculus
In this chapter, we give a brief introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play anExpand
Symmetries of the heat equation on the lattice
Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties areExpand
...
1
2
3
4
...