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) no. 1, pp. 1-27.

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

$${ϵ}_0=1,{ϵ}_{2n}={ϵ}_n,{ϵ}_{2n+1}=1-{ϵ}_n.$$

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

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

$${ϵ}_0=1,{ϵ}_{2n}={ϵ}_n,{ϵ}_{2n+1}=1-{ϵ}_n.$$

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

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