An algorithm to compute Liouvillian solutions of prime order linear difference–differential equations

A normal form is given for integrable linear difference–differential equations {σ(Y)=AY,δ(Y)=BY}, which is irreducible over C(x,t) and solvable in terms of Liouvillian solutions. We refine this normal form and devise an algorithm to compute all Liouvillian solutions of such kinds of systems of prime order.