Prouhet–Thue–Morse constant

In mathematics and its applications, the Prouhet-Thue-Morse constant is the number $au$ whose binary expansion .01101001100101101001011001101001... is given by the Prouhet-Thue-Morse sequence. That is,: $au\; =\; sum\_\{i=0\}^\{infty\}\; frac\{t\_i\}\{2^\{i+1\; =\; 0.412454033640\; ldots$

where $t\_i$ is the $i$-th element of the Prouhet-Thue-Morse sequence.

The generating series for the $t\_i$ is given by :$au(x)\; =\; sum\_\{i=0\}^\{infty\}\; (-1)^\{t\_i\}\; ,\; x^i\; =\; frac\{1\}\{1-x\}\; -\; 2\; sum\_\{i=0\}^\{infty\}\; t\_i\; ,\; x^i$and can be expressed as

: $au(x)\; =\; prod\_\{n=0\}^\{infty\}\; (\; 1\; -\; x^\{2^n\}\; ).$Note curiously that this is the product of Frobenius polynomials, and thus generalizes to arbitrary fields.

This number was shown to be transcendental by Kurt Mahler in 1929. [Kurt Mahler, "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen", "Math. Annalen", t. 101 (1929), p. 342–366.]

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 can be easily understood due to the nature of period doubling in the Mandelbrot set.

References

External links

* [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A010060 On-Line Encyclopedia of Integer Sequences Entry A010060]
* [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]
* [http://www.linas.org/art-gallery/escape/phase/atlas.html Parameter Ray Atlas] (2000) provides a link to the Mandelbrot set.