# Prouhet–Thue–Morse constant

(Redirected from 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 ith 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.