This thesis looks at the interplay of three important domains: combinatorics on words, theory of finite-state automata and number theory. More precisely, we show how tools coming from combinatorics on words and theory of finite-state automata intervene in the study of arithmetical problems concerning the Laurent series with coefficients in a finite field.The starting point of this thesis is a famous theorem of Christol which characterizes algebraic Laurent series over the field F_q(T), q being a power of the prime number p, in terms of finite-state automata and whose statement is the following : “A Laurent series with coefficients in a finite field F_q is algebraic over F_q(T) if and only if the sequence of its coefficients is p-automatic”.This result, which reveals, somehow, the simplicity of these Laurent series, has given rise to important works including numerous applications and generalizations. The theory of finite-state automata and the combinatorics on words naturally occur in number theory and, sometimes, prove themselves to be indispensable in establishing certain important results in this domain.The main purpose of this thesis is, foremost, to exploit the simplicity of the algebraic Laurent series with coefficients in a finite field in order to obtain some Diophantine results, then to try to extend this study to some interesting transcendental functions. First, we focus on a particular set of algebraic Laurent series that generalize the famous cubic introduced by Baum and Sweet. The main result we obtain concerning these Laurent series gives the explicit description of its continued fraction expansion, generalizing therefore some articles of Mills and Robbins.Unfortunately, it is often very difficult to find the continued fraction representation of a Laurent series, whether it is given by an algebraic equation or by its Laurent series expansion. The second study that we present in this thesis provides a Diophantine information which, although a priori less complete than the continued fraction expansion, has the advantage to characterize any algebraic Laurent series. The main idea is to use some the automaticity of the sequence of coefficients of these Laurent series in order to obtain a general bound for their irrationality exponent. In the last part of this thesis we focus on a more general class of Laurent series, namely the one of Laurent series of “low” complexity. We prove that this set includes some interesting functions, as for example the algebraic series or the inverse of the analogue of the real number \pi. We also show that this set satisfy some nice closure properties : in particular, it is a vector space over the field over F_q(T).
