Hankel determinants of the Thue-Morse sequence. (English) Zbl 0974.11010

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

Ann. Inst. Fourier 48, No. 1, 1-27 (1998).

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

Keywords:

Thue-Morse sequence; period doubling sequence; automatic sequences; Hankel determinants; Padé approximants

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