Algebraic independence of certain Mahler functions

We prove algebraic independence of functions satisfying a simple form of algebraic Mahler functional equations. The main result (Theorem 1.1) partly generalizes a result obtained by Kubota. This result is deduced from a quantitative version of it (Theorem 2.1), which is proved by using an inductive method originated by Duverney. As an application we can also generalize a recent result by Bundschuh and the second named author (Theorem 1.2 and its corollary).