Lacunary formal power series and the Stern-Brocot sequence

• Autorzy: Jean-Paul Allouche , Michel Mendès France
• Języki publikacji: EN

Abstrakty

EN

Let $F(X) = ∑_{n ≥ 0} (-1)^{εₙ} X^{-λₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and $λ_{n+1}/λₙ > 2$. 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 properties in the special case $λₙ = 2^{n+1} - 1$.

Kategorie tematyczne

• 13F25: Formal power series rings
• 11B85: Automata sequences
• 11A55: Continued fractions
• 11B83: Special sequences and polynomials
• 11B65: Binomial coefficients; factorials; q -identities

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 1
• Strony: 47-61

Daty

wydano

2013

Twórcy

autor

Jean-Paul Allouche

• Équipe Combinatoire et Optimisation, CNRS, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, F-75252 Paris Cedex 05, France

autor

Michel Mendès France

• Mathématiques, Université Bordeaux I, F-33405 Talence Cedex, France

Identyfikatory

DOI

10.4064/aa159-1-3