k-Regular Power Series and Mahler-Type Functional Equations

✍ Scribed by P.G. Becker

• Book ID: 102973422
• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 654 KB
• Volume: 49
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1994.1093

✦ Synopsis

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 (k_{1})-regular and (k_{2})-regular for multiplicatively independent (k_{1}) and (k_{2}) has to be a rational function. We note that this conjecture is a special case of a conjecture of Loxton and van der Poorten. C 1994 Academic Press, Inc.