A direct algorithm to compute rational solutions of first order linear \(q\)-difference systems (Q1348136)

A system of first order \(q\)-difference equations with rational coefficients over a field \(K\) of characteristic zero is considered. It is assumed that coefficients of the system are polynomials over the field \(K\). It is shown that by combining differential and difference approaches it is possible to completely solve the problem of computing all rational solutions of the \(q\)-difference equations system. The solution proceeds in two steps. First a universal denominator \(U(x)\) is constructed and next using the substitition \[ y(x)=z(x)/U(x) \] into the system the problem is reduced to finding polynomial solutions of the same type system in \(z(x)\).