Hankel determinants of the Thue-Morse sequence
Allouche, Jean-Paul ; Peyrière, Jacques ; Wen, Zhi-Xiong ; Wen, Zhi-Ying
Annales de l'Institut Fourier, Tome 48 (1998), p. 1-27 / Harvested from Numdam
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Soit ϵ=(ϵ n ) n≥0 la suite de Thue-Morse, c’est-à-dire la suite définie par les relations de récurrence :

ϵ0=1,ϵ2n=ϵn,ϵ2n+1=1-ϵn.

Soit {|ℰ n p |} n≥1,p≥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 : ∑ n≥0 (-1) ϵ n x n .

Let ϵ=(ϵ n ) n≥0 be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations:

ϵ0=1,ϵ2n=ϵn,ϵ2n+1=1-ϵn.

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 .

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     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},
     volume = {48},
     year = {1998},
     pages = {1-27},
     doi = {10.5802/aif.1609},
     mrnumber = {99a:11024},
     zbl = {0974.11010},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 48 (1998) pp. 1-27. doi : 10.5802/aif.1609. http://gdmltest.u-ga.fr/item/AIF_1998__48_1_1_0/

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