Embree-Trefethen constant

The Embree-Trefethen constant is a mathematical constant . It is a limit coefficient in number theory and is denoted by β * .

For a fixed real β consider the recursion

x _{n + 1} = x _n ± β x _{n - 1}

where the sign in the sum is chosen independently for each n with the same probability as '+' or '-'.

For β = 1 we get the random Fibonacci sequence .

It can be shown that for any β the limit value

${\displaystyle \sigma (\beta )=\lim _{n\to \infty }\left(|x_{n}|^{\frac {1}{n}}\right)}$

almost certainly exists. In other words: The sequence behaves asymptotically exponentially with probability 1 with base σ ( β ).

It applies

σ <1 for 0 < β < β * ≈ 0.70258,

so the sequence of x _n almost certainly falls asymptotically exponentially, and

σ > 1 for β > β *

so the terms of the sequence will almost certainly grow asymptotically exponentially.

Special values ​​of σ are:

• σ (1) = 1.13198 82487 943… ( Viswanath constant ) and
• σ ( β * ) = 1.

literature

• Mark Embree, Lloyd Nicholas Trefethen : Growth and decay of random Fibonacci sequences , Proceedings of the Royal Society A 455, July 1999, pp. 2471–2485 (English)