Algebraic independence and linear difference equations

We consider pairs of automorphisms (\phi,\sigma) acting on fields of Laurent or Puiseux series: pairs of shift operators (\phi\colon x\mapsto x+h_1, \sigma\colon x\mapsto x+h_2) , of q -difference operators (\phi\colon x\mapsto q_1x , \sigma\colon x\mapsto q_2x) , and of Mahler operators (\phi\colon x\mapsto x^{p_1},\ \sigma\colon x\mapsto x^{p_2}) . Given a solution f to a linear \phi -equation and a solution g to an algebraic \sigma -equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q -hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the \sigma -Galois theory of linear \phi -equations.