On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters

We present algorithms to compute the differential Galois group G associated via the parameterized Picard–Vessiot theory to a parameterized second-order linear differential equation∂2∂x2Y+r1∂∂xY+r0Y=0, where the coefficients r1r1 and r0r0 belong to the field of rational functions F(x)F(x) over a computable Π-field F of characteristic zero, and the finite set of commuting derivations Π is thought of as consisting of derivations with respect to parameters. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G   under the assumption that r1=0r1=0, which guarantees that G   is unimodular. When r1≠0r1≠0, we reinterpret a classical change-of-variables procedure in Galois-theoretic terms in order to reduce the computation of G to the computation of an associated unimodular differential Galois group H  . We establish a parameterized version of the Kolchin–Ostrowski theorem and apply it to give more direct proofs than those found in the literature of the fact that the required computations can be performed effectively. We then extract from these algorithms a complete set of criteria to decide whether any of the solutions to a parameterized second-order linear differential equation is Π-transcendental over the underlying Π-field of F(x)F(x). We give various examples of computation and some applications to differential transcendence.