Prouhet–Thue–Morse constant
In
where tn is the nth element of the Prouhet–Thue–Morse sequence.
Other representations
The Prouhet–Thue–Morse constant can also be expressed, without using tn , as an infinite product,[1]
This formula is obtained by substituting x = 1/2 into generating series for tn
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.[2]
Transcendence
The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.[3]
He also showed that the number
is also transcendental for any algebraic number α, where 0 < |α| < 1.
Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an
Appearances
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 [5]
See also
- Euler-Mascheroni constant
- Fibonacci word
- Golay–Rudin–Shapiro sequence
- Komornik–Loreti constant
Notes
- ^ Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
- ^ Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3).
- S2CID 120549929.
- doi:10.5802/aif.2666.
- ^ Allouche, Jean-Paul; Shallit, Jeffrey (1999). "The Ubiquitous Prouhet-Thue-Morse Sequence". Discrete Mathematics and Theoretical Computer Science: 11.
References
- Allouche, Jean-Paul; Zbl 1086.11015..
- Pytheas Fogg, N. (2002). Zbl 1014.11015.
External links
- OEIS sequence A010060 (Thue-Morse sequence)
- The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
- PlanetMath entry