Hankel determinants of the Thue-Morse sequence

Jean-Paul Allouche; Jacques Peyrière; Zhi-Xiong Wen; Zhi-Ying Wen

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 1, page 1-27
  • ISSN: 0373-0956

Abstract

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Let ϵ = ( ϵ n ) n ≥ 0 be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations: ϵ 0 = 1 , ϵ 2 n = ϵ n , ϵ 2 n + 1 = 1 - ϵ n . We consider { | ℰ n p | } n ≥ 1 , p ≥ 0 , the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series ∑ n ≥ 0 ( - 1 ) ϵ n x n .

How to cite

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Allouche, Jean-Paul, et al. "Hankel determinants of the Thue-Morse sequence." Annales de l'institut Fourier 48.1 (1998): 1-27. <http://eudml.org/doc/75275>.

@article{Allouche1998,
abstract = {Let $\epsilon =(\epsilon _n)_\{n\ge 0\}$ be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations:\begin\{\}\epsilon \_0=1,~\epsilon \_\{2n\}=\epsilon \_n,~\epsilon \_\{2n+1\}=1-\epsilon \_n.\end\{\}We consider $\lbrace \vert \{\cal E\}^p_n\vert \rbrace _\{n\ge 1,p\ge 0\}$, the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series $\sum _\{n\ge 0\}(-1)^\{\epsilon _n\}x^n$.},
author = {Allouche, Jean-Paul, Peyrière, Jacques, Wen, Zhi-Xiong, Wen, Zhi-Ying},
journal = {Annales de l'institut Fourier},
keywords = {Thue-Morse sequence; period doubling sequence; automatic sequences; Hankel determinants; Padé approximants},
language = {eng},
number = {1},
pages = {1-27},
publisher = {Association des Annales de l'Institut Fourier},
title = {Hankel determinants of the Thue-Morse sequence},
url = {http://eudml.org/doc/75275},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Allouche, Jean-Paul
AU - Peyrière, Jacques
AU - Wen, Zhi-Xiong
AU - Wen, Zhi-Ying
TI - Hankel determinants of the Thue-Morse sequence
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 1
SP - 1
EP - 27
AB - Let $\epsilon =(\epsilon _n)_{n\ge 0}$ be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations:\begin{}\epsilon _0=1,~\epsilon _{2n}=\epsilon _n,~\epsilon _{2n+1}=1-\epsilon _n.\end{}We consider $\lbrace \vert {\cal E}^p_n\vert \rbrace _{n\ge 1,p\ge 0}$, the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series $\sum _{n\ge 0}(-1)^{\epsilon _n}x^n$.
LA - eng
KW - Thue-Morse sequence; period doubling sequence; automatic sequences; Hankel determinants; Padé approximants
UR - http://eudml.org/doc/75275
ER -

References

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