Algebraic independence of functions satisfying certain Mahler type functional equations and its applications

One of the techniques used to prove the algebraic independence of numbers is Mahler’s method, which deals with the values of so-called Mahler functions satisfying a certain type of functional equation. In order to apply the method, one must confirm the algebraic independence of the Mahler functions themselves. This can be reduced, in many cases, to their linear independence modulo the rational function field, that is, the problem of determining whether a nonzero linear combination of them is a rational function or not. In the case of one variable, this can be treated by arguments involving poles of rational functions. However, in the case of several variables, this method is not available. In this paper we shall resolve this problem by considering a generic point of an irreducible algebraic variety. Theorems 1 and 2 in this paper assert that certain types of functional equations in several variables have no nontrivial rational function solutions. As applications, we shall prove the algebraic independence of various kinds of reciprocal sums of linear recurrences in Theorems 3, 4, and 5, and that of the values at algebraic numbers of power series, Lambert series, and infinite products generated by linear recurrences in Theorem 6. Let Ω = (ωij) be an n × n matrix with nonnegative integer entries. If z = (z1, . . . , zn) is a point of C n with C the set of complex numbers, we define a transformation Ω : C → C by