# 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 ${\displaystyle \tau }$—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 ${\displaystyle t_{i}}$ is the ith element of the Prouhet–Thue–Morse sequence.

The generating series for the ${\displaystyle t_{i}}$ 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]

## 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.