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In [[mathematics]], the '''Prouhet–Thue–Morse constant''', named for [[Eugène Prouhet]], [[Axel Thue]], and [[Marston Morse]], is the number — denoted by <math>\tau</math> — whose [[binary expansion]] .01101001100101101001011001101001... is given by the [[Thue–Morse sequence]]. That is, | |||
: <math> \tau = \sum_{i=0}^{\infty} \frac{t_i}{2^{i+1}} = 0.412454033640 \ldots </math> | |||
where <math>t_i</math> is the ''i''<sup>th</sup> element of the Prouhet–Thue–Morse sequence. | |||
The generating series for the <math>t_i</math> is given by | |||
:<math> \tau(x) = \sum_{i=0}^{\infty} (-1)^{t_i} \, x^i = \frac{1}{1-x} - 2 \sum_{i=0}^{\infty} t_i \, x^i</math> | |||
and can be expressed as | |||
: <math> \tau(x) = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ). </math> | |||
This is the product of [[Frobenius polynomial]]s, and thus generalizes to arbitrary [[field (mathematics)|fields]]. | |||
The Prouhet–Thue–Morse constant was shown to be [[transcendental number|transcendental]] by [[Kurt Mahler]] in 1929.<ref>{{cite journal | first=Kurt | last=Mahler | authorlink=Kurt Mahler | title=Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen | journal=[[Math. Annalen]] | volume=101 | year=1929 | pages=342–366 | jfm=55.0115.01 }}</ref> | |||
== Applications == | |||
The Prouhet–Thue–Morse constant occurs as the angle of the [[External ray|Douady–Hubbard ray]] at the end of the sequence of western bulbs of the [[Mandelbrot set]]. This property is tied to the nature of [[period doubling]] in the Mandelbrot set.<ref>[http://www.linas.org/art-gallery/escape/phase/atlas.html Parameter Ray Atlas] (2000) provides a link to the Mandelbrot set.</ref>{{clarify|date=August 2010}} | |||
==Notes== | |||
<references /> | |||
==References== | |||
*{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}. | |||
* {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }} | |||
== External links == | |||
* {{SloanesRef |sequencenumber=A010060|name=Thue-Morse sequence}} | |||
* [http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps The ubiquitous Prouhet-Thue-Morse sequence], John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history | |||
* [http://planetmath.org/encyclopedia/ProuhetThueMorseConstant.html PlanetMath entry] | |||
{{DEFAULTSORT:Prouhet-Thue-Morse Constant}} | |||
[[Category:Mathematical constants]] | |||
[[Category:Number theory]] | |||
[[Category:Transcendental numbers]] | |||
Revision as of 13:38, 11 June 2013
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]
Applications
The Prouhet–Thue–Morse constant occurs as the angle of the Douady–Hubbard ray at the end of the sequence of western bulbs of the Mandelbrot set. This property is tied to the nature of period doubling in the Mandelbrot set.[2]Template:Clarify
Notes
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References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Template:SloanesRef
- The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
- PlanetMath entry