On solutions of linear ordinary difference equations in their coefficient field (Q1581126)

The paper is devoted to the problem, given a difference field \(k\) with an automorphism \(\sigma\), \(g\in k\) and a linear ordinary difference operator \(L\) with coefficients in \(k\), to compute all solutions in \(k\) of the equation \(Ly=g\). There are known solutions to this problem when \(k=C(x)\) and \(\sigma\) is the automorphism over \(C\) given by \(\sigma x=x+1\) or \(\sigma x=qx\), \(q\in C^{*}\) [\textit{S. A. Abramov}, Program. Comput. Softw. 21, 273-278 (1995; Zbl 0910.65107)]. The author extends the notion of monomial extensions of differential fields to difference fields and investigates such extensions. This yields in particular a new algorithm for computing the rational solutions of \(q\)-difference equations with polynomial coefficients.