Functional relations for solutions of q-difference equations

In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333-377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962-4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a q-difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its q-transforms. We apply our results to q-hypergeometric series.