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

Abstract

A normal form is given for integrable linear difference–differential equations { σ ( Y ) = A Y , δ ( Y ) = B Y } , 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.