Prouhet–Thue–Morse constant

In mathematics, the Prouhet–Thue–Morse constant, named for , Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence. That is,

${\displaystyle \tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}=0.412454033640\ldots }$

where t_i is the i^th element of the Prouhet–Thue–Morse sequence.

The generating series for the t_i is given by

${\displaystyle \tau (x)=\sum _{i=0}^{\infty }(-1)^{t_{i}}\,x^{i}={\frac {1}{1-x}}-2\sum _{i=0}^{\infty }t_{i}\,x^{i}}$

and can be expressed as

${\displaystyle \tau (x)=\prod _{n=0}^{\infty }(1-x^{2^{n}}).}$

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]

See also[]

• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Komornik–Loreti constant

Notes[]

References[]

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015..
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. 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

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