Prouhet–Thue–Morse constant

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

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

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

The generating series for the t_i is given by

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

and can be expressed as

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

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.

See also

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

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