ANNALES DE L'INSTITUT FOURIER

Hankel determinants of the Thue-Morse sequence
Annales de l'Institut Fourier, Volume 48 (1998) no. 1, pp. 1-27.

Let $ϵ=\left({ϵ}_{n}{\right)}_{n\ge 0}$ be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations:

 ${ϵ}_{0}=1,\phantom{\rule{3.33333pt}{0ex}}{ϵ}_{2n}={ϵ}_{n},\phantom{\rule{3.33333pt}{0ex}}{ϵ}_{2n+1}=1-{ϵ}_{n}.$

We consider $\left\{|{ℰ}_{n}^{p}|{\right\}}_{n\ge 1,p\ge 0}$, 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 ${\sum }_{n\ge 0}\left(-1{\right)}^{{ϵ}_{n}}{x}^{n}$.

Soit $ϵ=\left({ϵ}_{n}{\right)}_{n\ge 0}$ la suite de Thue-Morse, c’est-à-dire la suite définie par les relations de récurrence :

 ${ϵ}_{0}=1,\phantom{\rule{3.33333pt}{0ex}}{ϵ}_{2n}={ϵ}_{n},\phantom{\rule{3.33333pt}{0ex}}{ϵ}_{2n+1}=1-{ϵ}_{n}.$

Soit $\left\{|{ℰ}_{n}^{p}|{\right\}}_{n\ge 1,p\ge 0}$, la suite double des déterminants de Hankel (modulo 2) associés à la suite de Thue-Morse. Elle vérifie un ensemble complexe de relations de récurrence. On montre qu’elle est 2-automatique. On donne des applications, notamment à l’étude combinatoire de la suite de Thue-Morse et à l’existence de certains approximants de Padé de la série formelle : ${\sum }_{n\ge 0}\left(-1{\right)}^{{ϵ}_{n}}{x}^{n}$.

@article{AIF_1998__48_1_1_0,
author = {Allouche, Jean-Paul and Peyri\ere, Jacques and Wen, Zhi-Xiong and Wen, Zhi-Ying},
title = {Hankel determinants of the {Thue-Morse} sequence},
journal = {Annales de l'Institut Fourier},
pages = {1--27},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {48},
number = {1},
year = {1998},
doi = {10.5802/aif.1609},
zbl = {0974.11010},
mrnumber = {99a:11024},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1609/}
}
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
SP  - 1
EP  - 27
VL  - 48
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1609/
DO  - 10.5802/aif.1609
LA  - en
ID  - AIF_1998__48_1_1_0
ER  - 
%0 Journal Article
%A Allouche, Jean-Paul
%A Peyrière, Jacques
%A Wen, Zhi-Xiong
%A Wen, Zhi-Ying
%T Hankel determinants of the Thue-Morse sequence
%J Annales de l'Institut Fourier
%D 1998
%P 1-27
%V 48
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1609/
%R 10.5802/aif.1609
%G en
%F AIF_1998__48_1_1_0
Allouche, Jean-Paul; Peyrière, Jacques; Wen, Zhi-Xiong; Wen, Zhi-Ying. Hankel determinants of the Thue-Morse sequence. Annales de l'Institut Fourier, Volume 48 (1998) no. 1, pp. 1-27. doi : 10.5802/aif.1609. https://aif.centre-mersenne.org/articles/10.5802/aif.1609/`

[1] J.-P. Allouche, Automates finis en théorie des nombres, Expo. Math., 5 (1987), 239-266. | MR | Zbl

[2] G.A. Baker and Jr.P. Gravers-Morris, Padé approximants, Encyclopedia of mathematics and its applications, I, II, Cambridge University Press, 1981. | Zbl

[3] C. Brezinski, Padé-type approximation and general orthogonal polynomials, Birkhäuser Verlag, 1980. | MR | Zbl

[4] G. Christol, T. Kamae, M. Mendès France and G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. France, 108 (1980), 401-419. | Numdam | MR | Zbl

[5] A. Cobham, A proof of transcendence based on functional equations, IBM RC-2041, Yorktown Heights, New York, 1968.

[6] A. Cobham, Uniform tag sequences, Math. Systems Theory, 6 (1972), 164-192. | MR | Zbl

[7] F.M. Dekking, Combinatorial and statistical properties of sequences generated by substitutions, Thesis, Mathematisch Instituut, Katholieke Universiteit van Nijmegen, 1980.

[8] F.M. Dekking, M. Mendès France and A.J. Van Der Poorten, Folds!, Math. Intelligencer, 4 (1982), 130-138, 173-181 and 190-195. | Zbl

[9] W.H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc., 109 (1963), 467-491. | MR | Zbl

[10] M. Morse, Recurrent geodesic on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100. | JFM

[11] M. Queffélec, Substitution dynamical systems — Spectral analysis, Lecture Notes in Math., 1294, Springer-Verlag (1987). | MR | Zbl

[12] O. Salon, Suites automatiques à multi-indices et algébricité, C.R. Acad. Sci. Paris, Série I, 305 (1987), 501-504. | MR | Zbl

[13] O. Salon, Suites automatiques à multi-indices, Séminaire de Théorie des Nombres de Bordeaux, Exposé 4, (1986-1987), 4-01-4-27; followed by an appendix by J. Shallit, 4-29A-4-36A. | Zbl

[14] A. Thue, Über unendliche Zeichenreihen, Norske vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana, 7 (1906), 1-22. | JFM

[15] A. Thue, Über die gegenseitige Lage gleicher Teile gewisse Zeichenreihen, Norske vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana, 1 (1912), 1-67. | JFM

[16] Z.-X. Wen and Z.-Y. Wen, The sequences of substitutions and related topics, Adv. Math. China, 3 (1989), 123-145. | Zbl

[17] Z.-X. Wen and Z.-Y. Wen, Mots infinis et produits de matrices à coefficients polynomiaux, RAIRO, Theoretical Informatics and Applications, 26 (1992), 319-343. | Numdam | MR | Zbl

[18] Z.-X. Wen and Z.-Y. Wen, Some studies on the (p,q)-type sequences, Theoret. Comput. Sci., 94 (1992), 373-393. | MR | Zbl

Cited by Sources: