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Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

Let F ( X ) = n 0 ( - 1 ) ε X - λ be a real lacunary formal power series, where εₙ = 0,1 and λ n + 1 / λ > 2 . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that Q ω ( X ) is a polynomial if and only if ω ∈ ℤ. In all the other cases Q ω ( X ) is an infinite formal power series; we discuss its algebraic...

Langage de Łukasiewicz et diagonales de séries formelles

Isabelle Fagnot (1996)

Journal de théorie des nombres de Bordeaux

Dans un corps fini, toute série formelle algébrique en une indéterminée est la diagonale d'une fraction rationnelle en deux indéterminées (Furstenberg 67). Dans cet article, nous donnons une nouvelle preuve de ce résultat, par des méthodes purement combinatoires.

Languages under substitutions and balanced words

Alex Heinis (2004)

Journal de Théorie des Nombres de Bordeaux

This paper consists of three parts. In the first part we prove a general theorem on the image of a language K under a substitution, in the second we apply this to the special case when K is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth.

Logarithmic frequency in morphic sequences

Jason P. Bell (2008)

Journal de Théorie des Nombres de Bordeaux

We study the logarithmic frequency of letters and words in morphic sequences and show that this frequency must always exist, answering a question of Allouche and Shallit.

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