K-Regular Power Series and Mahler-Type Functional Equations

Abstract

Allouche and Shallit generalized the concept of k-automatic sequences by introducing the notion of k-regular sequences and k-regular power series. We show that k-regular power series satisfy Mahler-type functional equations, and that power series satisfying Mahler-type functional equations of a somewhat special type must be k-regular. This generalizes earlier work of Christol et al. As an application we deduce transcendence results for the values of k-regular power series at algebraic points, thus answering a question of Allouche and Shallit. We also show how Mahler-type functional equations lead to transcendence results in the case of power series with coefficients from a finite field. This generalizes earlier work of Wade and results of Allouche. Allouche and Shallit conjectured that a power series which is k1-regular and k2-regular for multiplicatively independent k1 and k2 has to be a rational function. We note that this conjecture is a special case of a conjecture of Loxton and van der Poorten.