In mathematics, the Prouhet–Thue–Morse constant, named for, Axel Thue, and Marston Morse, is the number—denoted by —whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,
\tau=
| infty | |
| \sum | |
| n=0 |
| tn | |
| 2n+1 |
=0.412454033640\ldots
The Prouhet–Thue–Morse constant can also be expressed, without using, as an infinite product,
\tau=
| 1 | |
| 4 |
| infty | ||
| \left[2-\prod | \left(1- | |
| n=0 |
| 1 | ||||
|
\right)\right]
This formula is obtained by substituting x = 1/2 into generating series for
F(x)=
| infty | |
| \sum | |
| n=0 |
| tn | |
| (-1) |
xn=
| infty | |
| \prod | |
| n=0 |
(1-
| 2n | |
| x |
)
The continued fraction expansion of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]
Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.[1]
The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.[2]
He also showed that the number
| infty | |
| \sum | |
| i=0 |
tn\alphan
Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an irrationality measure of 2.[3]
The Prouhet–Thue–Morse constant appears in probability. If a language L over is chosen at random, by flipping a fair coin to decide whether each word w is in L, the probability that it contains at least one word for each possible length is [4]
p=
| infty | ||
| \prod | \left(1- | |
| n=0 |
| 1 | ||||
|
\right)=
| infty | |
| \sum | |
| n=0 |
| |||||
| 2n+1 |
=2-4\tau=0.35018386544\ldots