  We apply the difference-differential Galois theory developed by Hardouin and
Singer to compute the differential-algebraic relations among the solutions to a
second-order homogeneous linear difference equation of the form $
y(x+2)+a(x)y(x+1)+b(x)y(x)=0,$ where the coefficients $a(x),b(x)\in
\bar{\mathbb{Q}}(x)$ are rational functions in $x$ with coefficients in
$\bar{\mathbb{Q}}$. We develop algorithms to compute the
difference-differential Galois group associated to such an equation, and show
how to deduce the differential-algebraic relations among the solutions from the
defining equations of the Galois group.
