Prouhet-Thue-Morse constant

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In mathematics and its applications, the Prouhet-Thue-Morse constant is the number $τ$ whose binary expansion is the Prouhet-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

$\tau(x) = \sum_{i=0}^{\infty} (-1)^{t_i} \, x^i = \frac{1}{1-x} - \sum_{i=0}^{\infty} t_i \, x^i$

and can be expressed as

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

Note curiously that this is the product of Frobenius polynomials, and thus generalizes to arbitrary fields.

This number has been shown to be transcendental by K. Mahler in 1929.

[edit] Applications

The Prouhet-Thue-Morse constant occurs in a number of mathematical contexts. Some of these are listed below.

It occurs as the angle of the Douady-Hubbard ray at the end of the sequence of western bulbs of the Mandelbrot set. This can be easily understood due to the nature of period doubling in the Mandelbrot set.