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 —whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence. That is,
where is the ith element of the Prouhet–Thue–Morse sequence.
The generating series for the is given by
and can be expressed as
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.
References
- 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.
External links
- "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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