Algebraic independence and linear difference equations

Abstract

We consider pairs of automorphisms $(\ddot{I}•,\ddot{I}ƒ)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(\ddot{I}•:x\mapsto x+h_1,\ddot{I}ƒ:x\mapsto x+h_2)$, of $q$-difference operators $(\ddot{I}•:x\mapsto q_{1}x,\ddot{I}ƒ:x\mapsto q_{2}x)$, and of Mahler operators $(\ddot{I}•:x\mapsto x^{p_1},\ddot{I}ƒ:x\mapsto x^{p_2})$. Given a solution $f$ to a linear $\ddot{I}•$-equation and a solution $g$ to a linear $\ddot{I}ƒ$-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 $\ddot{I}ƒ$-Galois theory of linear $\ddot{I}•$-equations.