Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, and Michel Mendès France. "Lacunary formal power series and the Stern-Brocot sequence." Acta Arithmetica 159.1 (2013): 47-61. <http://eudml.org/doc/279381>.

@article{Jean2013,
abstract = {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 $λₙ = 2^\{n+1\} - 1$.},
author = {Jean-Paul Allouche, Michel Mendès France},
journal = {Acta Arithmetica},
keywords = {Stern-Brocot sequence; continued fractions of formal power series; automatic sequences; algebraicity of formal power series},
language = {eng},
number = {1},
pages = {47-61},
title = {Lacunary formal power series and the Stern-Brocot sequence},
url = {http://eudml.org/doc/279381},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Jean-Paul Allouche
AU - Michel Mendès France
TI - Lacunary formal power series and the Stern-Brocot sequence
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 1
SP - 47
EP - 61
AB - 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 $λₙ = 2^{n+1} - 1$.
LA - eng
KW - Stern-Brocot sequence; continued fractions of formal power series; automatic sequences; algebraicity of formal power series
UR - http://eudml.org/doc/279381
ER -