Prouhet–Thue–Morse constant

In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by $\tau$ —whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence. That is,

\tau =\sum _{{i=0}}^{{\infty }}{\frac {t_{i}}{2^{{i+1}}}}=0.412454033640\ldots

where $t_{i}$ is the i^th element of the Prouhet–Thue–Morse sequence.

The generating series for the $t_{i}$ is given by

\tau (x)=\sum _{{i=0}}^{{\infty }}(-1)^{{t_{i}}}\,x^{i}={\frac {1}{1-x}}-2\sum _{{i=0}}^{{\infty }}t_{i}\,x^{i}

and can be expressed as

\tau (x)=\prod _{{n=0}}^{{\infty }}(1-x^{{2^{n}}}).

This is the product of Frobenius polynomials, and thus generalizes to arbitrary fields.

The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.^[1]

Notes

↑ Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01.

Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015. .
Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.

"Sloane's A010060 : Thue-Morse sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
PlanetMath entry

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