\(k\)-regular power series and Mahler-type functional equations (Q1342381)

The author shows that a \(k\)-regular power series satisfies a Mahler-type functional equation (the notion of \(k\)-regular power series has been introduced by Allouche and Shallit generalizing the concept of \(k\)- automatic power series). The author shows that the converse of this result is not true unless we add an extra hypothesis on the Mahler-type functional equation. Furthermore, from this property transcendence results at algebraic points can be deduced for \(k\)-regular power series with coefficients in the algebraic closure of \(\mathbb{Q}\) and also a generalization of a result of L. I. Wade (1944) (also shown by Allouche in 1990) concerning power series with coefficients in a finite field.