We consider pairs of automorphisms (φ, σ) acting on fields of Laurent or Puiseux series: pairs of shift operators (φ : x → x + h1, σ : x → x + h2), of q-difference operators (φ : x → q1x, σ : x → q2x), and of Mahler operators (φ : x → x p 1 , σ : x → x p 2). Given a solution f to a linear φ-equation and a solution g to a linear σ-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 σ-Galois theory of linear φ-equations.