Liouvillian Solutions of Difference-Differential Equations

Abstract: For a field k$with an automorphism \sigma and a derivation \delta, we introduce the notion of liouvillian solutions of linear difference-differential systems {\sigma(Y) = AY, \delta(Y) = BY} over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x,t), \sigma(x) = x+1, \delta = d/dt$ and the size of the system is a prime.