Prouhet–Thue–Morse constant

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In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet [fr], Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,

where tn is the nth element of the Prouhet–Thue–Morse sequence.

Other representations

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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

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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 irrationality measure of 2.[4]

Appearances

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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

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Notes

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  1. ^ Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
  2. ^ Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3).
  3. ^ Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929.
  4. ^ Bugaeud, Yann (2011). "On the rational approximation to the Thue–Morse–Mahler numbers". Annales de l'Institut Fourier. 61 (5): 2065–2076. doi:10.5802/aif.2666.
  5. ^ Allouche, Jean-Paul; Shallit, Jeffrey (1999). "The Ubiquitous Prouhet-Thue-Morse Sequence". Discrete Mathematics and Theoretical Computer Science: 11.

References

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