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 $\tau$—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.^[1]

Notes

1. Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01.

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). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.

External links

• "Sloane's A010060 : Thue-Morse sequence", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
• PlanetMath entry