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Hankel determinants of the ThueMorse sequence J.P. Allouche \Lambda J. Peyri`ere y Z.X. Wen z Z.Y. Wen x

Summary: Hankel determinants of the Thue­Morse sequence
J.­P. Allouche \Lambda J. Peyri`ere y Z.­X. Wen z Z.­Y. Wen x
November 5, 1996
Let ffl = (ffl n ) n–0 be the Thue­Morse sequence, i.e., the sequence defined by the
recurrence equations:
ffl 0 = 1; ffl 2n = ffl n ; ffl 2n+1 = 1 \Gamma ffl n :
We consider fjE p
n jg n–1;p–0 , the double sequence of Hankel determinants (modulo 2)
associated with the Thue­Morse sequence. Together with three other sequences, it
obeys a set of sixteen recurrence equations. It shown to be automatic. Applications
are given, namely to combinatorial properties of the Thue­Morse sequence and to the
existence of certain Pad'e approximants of the power series
(\Gamma1) ffl n x
n .
0 Introduction
Let S = fa; bg be a two­letter alphabet and S \Lambda the free monoid generated by S. Consider
the endomorphism ` defined on S \Lambda by


Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11


Collections: Computer Technologies and Information Sciences