Algebraic independence and linear difference equations

We consider pairs of automorphisms(Φσ) acting on fields of Laurent or Puiseux series: pairs of shift operators(Φ x →h1;σx +x h2) of q-difference operators(Φ x → q1x,σ x → q2x) and of Mahler operators (σ x → xp1 ; σ x → xp2 ) Given a solution f to a linear Φ-equation and a solution g to an algebraic Φ-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 Φ-Galois theory of linear Φ-equations.