Prouhet–Thue–Morse constant
In mathematics, the Prouhet–Thue–Morse constant, named for Template:Ill, Axel Thue, and Marston Morse, is the number—denoted by Template:Mvar—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,
- <math> \tau = \sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}} = 0.412454033640 \ldots </math>
where Template:Math is the Template:Math element of the Prouhet–Thue–Morse sequence.
Other representationsEdit
The Prouhet–Thue–Morse constant can also be expressed, without using Template:Math , as an infinite product,<ref name="mw">Template:Mathworld</ref>
- <math> \tau = \frac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right)\right] </math>
This formula is obtained by substituting x = 1/2 into generating series for Template:Math
- <math> F(x) = \sum_{n=0}^{\infty} (-1)^{t_n} x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ) </math>
The continued fraction expansion of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] (sequence A014572 in the OEIS)
Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.<ref>Template:Cite journal</ref>
TranscendenceEdit
The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.<ref>Template:Cite journal</ref>
He also showed that the number
- <math>\sum_{i=0}^{\infty} t_n \, \alpha^n</math>
is also transcendental for any algebraic number α, where 0 < |α| < 1.
Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2.<ref>Template:Cite journal</ref>
AppearancesEdit
The Prouhet–Thue–Morse constant appears in probability. If a language L over {0, 1} is chosen at random, by flipping a fair coin to decide whether each word w is in L, the probability that it contains at least one word for each possible length is <ref>Template:Cite journal</ref>
- <math> p = \prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right) = \sum_{n=0}^{\infty} \frac{(-1)^{t_n}}{2^{n+1}} = 2 - 4 \tau = 0.35018386544\ldots</math>
See alsoEdit
NotesEdit
<references />
ReferencesEdit
External linksEdit
- Template:OEIS el
- The ubiquitous Prouhet–Thue–Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
- PlanetMath entry