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: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalize...

Several explicit examples of quasi exactly solvable ‘discrete’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/ sin x potential de...

We extend and give short proofs of some recent results regarding some classes of rational difference equations.

We continue the study of algebraic difference equations of the type un+2un = ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2un = ψ(un+1) in R∗, related to families of conics or cubics: generalization of the Lyness’ sequences” (2004),...

In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based Taylor formula for scale, investigate difference transform method (DTM). This has been applied successfully to solve Riccati type equations in quantum calculus. To demonstrate ability and efficacy method, some examples have provided.

Acknowledgements First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master's thesis. I can not give enough thanks to his heartwarming encouragement all...

The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well as to their dependence on the derivative discretization. The case of q–symmetries for discrete equations in a q–lattice is also considered.