Abstract: Given a linear difference equation, there is a difference-differential Galois group that encodes the differential-algebraic dependencies among the solutions of the equation. After giving a brief introduction to this theory, I will describe algorithms to compute the Galois group associated to a second-order linear difference equation over C(x), the field of rational functions over a computable field C of characteristic zero, with respect to the C-linear shift automorphism that sends x to x+1. I will also discuss some concrete examples to illustrate these algorithms, and show explicitly in the examples how to derive the differential-algebraic dependencies among the solutions from the knowledge of the defining equations for the Galois group.