×

zbMATH — the first resource for mathematics

Hankel determinants of the Thue-Morse sequence. (English) Zbl 0974.11010
Summary: Let \(\varepsilon=(\varepsilon_n)_{n\geq 0}\) be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations: \[ \varepsilon_0=1, \varepsilon_{2n}=\varepsilon_n, \varepsilon_{2n+1}=1-\varepsilon_n. \] We consider \(\{|{\mathcal E}^p_n|\}_{n\geq 1,p\geq 0}\), 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 \(\sum_{n\geq 0}(-1)^{\varepsilon_n}x^n\).

MSC:
11B85 Automata sequences
68R15 Combinatorics on words
41A21 Padé approximation
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] J.-P. ALLOUCHE, Automates finis en théorie des nombres, Expo. Math., 5 (1987), 239-266. · 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. · 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. · 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. · 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. · 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). · Zbl 0642.28013
[12] O. SALON, Suites automatiques à multi-indices et algébricité, C.R. Acad. Sci. Paris, Série I, 305 (1987), 501-504. · 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. · 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. · Zbl 0758.11017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.