Decimal expansions of classical constants such as √2, π and ζ(3) have long been a source of difficult questions. In the case of Laurent series with coefficients in a finite field, where “no carries appear,” the situation seems to be simplified and drastically different. In 1935 Carlitz introduced analogs of real numbers such as π, e or ζ(3) and it became reasonable to enquire how “complex” the Laurent representation of these “numbers” is. In this paper we prove that the inverse of Carlitz’s analog of π, Πq, has, in general, linear subword complexity, except in the case q = 2, when the complexity is quadratic. In particular, this gives a new proof of the transcendence of Π2 over F2(T). In the second part, we consider the classes of Laurent series of at most polynomial complexity and of zero entropy. We show that these satisfy some nice closure properties.
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