Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

Abstract

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 [Formula: see text] where the coefficients [Formula: see text] are rational functions in [Formula: see text] with coefficients in [Formula: see text]. 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.