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 \(\mathbf {q}\)-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 \(\mathbf {q}\)-transforms. We apply our results to q-hypergeometric series.

在本文中，我们研究了q差分方程解及其相对于辅助算子的变换所满足的代数关系。我们的主要工具是在Hardouin和Singer（Math Ann 342（2）：333-377，2008）和Ovchinnikov和Wibmer（Int Math Res Not 12：3962-4018，2015）中开发的参数化Galois理论。本文的第一部分涉及辅助算子是一个导数的情况，而第二部分则处理\（\ mathbf {q} \）-差分算子。在这两种情况下，我们都提供了标准来保证级数的代数独立性，q差分方程的解及其连续的导数或其\（\ mathbf {q} \）变换。我们将结果应用于q-超几何级数。