Abstract
We present a survey of results concerning regular sequences and related objects. Regular sequences were defined in the early 1990s by Allouche and Shallit as a combinatorially, algebraically, and analytically interesting generalization of automatic sequences. In this chapter, after an historical introduction, we follow the development from automatic sequences to regular sequences, and their associated generating functions, to Mahler functions. We then examine size and growth properties of regular sequences. The last half of the chapter focuses on the algebraic, analytic, and Diophantine properties of Mahler functions. In particular, we survey the rational-transcendental dichotomies of Mahler functions, due to Bézivin, and of regular numbers, due to Bell, Bugeaud, and Coons.
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