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,

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

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

The generating series for the t_i is given by

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

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]