Applications of Lie groups to second-order discrete delay equations

This paper is concerned with applications of Lie groups to second-order discrete delay equation, which is similar to ordinary differential equations, but their evolution involves past values of the state variable. However Lie's methods cannot be directly applied to discrete delay equation. To apply the Lie group, it is not only necessary to consider the current state, but also the state at previous moments. We consider the representation of Lie group with respect to a change of the discrete dependent and independent variables, and then use the found invariants to form a discrete delay equation. As an application, we also give some examples to demonstrate our results.