Hankel determinants of the Thue-Morse sequence
Annales de l'Institut Fourier, Volume 48 (1998) no. 1, p. 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},
publisher = {Association des Annales de l'institut Fourier},
volume = {48},
number = {1},
year = {1998},
pages = {1-27},
doi = {10.5802/aif.1609},
zbl = {0974.11010},
mrnumber = {99a:11024},
language = {en},
url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_1998__48_1_1_0/`

[1] J.-P. Allouche, Automates finis en théorie des nombres, Expo. Math., 5 (1987), 239-266. | MR 88k:11046 | Zbl 0641.10041

[2] G.A. Baker and Jr.P. Gravers-Morris, Padé approximants, Encyclopedia of mathematics and its applications, I, II, Cambridge University Press, 1981. | Zbl 0603.30044

[3] C. Brezinski, Padé-type approximation and general orthogonal polynomials, Birkhäuser Verlag, 1980. | MR 82a:41017 | Zbl 0418.41012

[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 82e:10092 | Zbl 0472.10035

[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 56 #15230 | Zbl 0253.02029

[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 0493.10001

[9] W.H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc., 109 (1963), 467-491. | MR 32 #8325 | Zbl 0121.18002

[10] M. Morse, Recurrent geodesic on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100. | JFM 48.0786.06

[11] M. Queffélec, Substitution dynamical systems — Spectral analysis, Lecture Notes in Math., 1294, Springer-Verlag (1987). | MR 89g:54094 | Zbl 0642.28013

[12] O. Salon, Suites automatiques à multi-indices et algébricité, C.R. Acad. Sci. Paris, Série I, 305 (1987), 501-504. | MR 88k:11094 | Zbl 0628.10007

[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 0653.10049

[14] A. Thue, Über unendliche Zeichenreihen, Norske vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana, 7 (1906), 1-22. | JFM 39.0283.01

[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 44.0462.01

[16] Z.-X. Wen and Z.-Y. Wen, The sequences of substitutions and related topics, Adv. Math. China, 3 (1989), 123-145. | Zbl 0694.10006

[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 93d:15035 | Zbl 0758.11016

[18] Z.-X. Wen and Z.-Y. Wen, Some studies on the (p,q)-type sequences, Theoret. Comput. Sci., 94 (1992), 373-393. | MR 93g:11022 | Zbl 0758.11017