Somos' quadratic recurrence constant

In mathematical analysis and number theory, Somos' quadratic recurrence constant or simply Somos' constant is a constant defined as an expression of infinitely many nested square roots. It arises when studying the asymptotic behaviour of a certain sequence The constant named after Michael Somos. It is defined by:

:<math>\sigma = \sqrt {1 \sqrt {2 \sqrt{3 \sqrt{4\sqrt{5\cdots}</math>

which gives a numerical value of approximately:

:<math>\sigma = 1.661687949633594121295\dots\;</math> .

Sums and products

Somos' constant can be alternatively defined via the following infinite product:

:<math>\sigma=\prod_{k=1}^\infty k^{1/2^k} =

1^{1/2}\;2^{1/4}\; 3^{1/8}\; 4^{1/16} \dots</math>

This can be easily rewritten into the far more quickly converging product representation

:<math>\sigma =

\left(\frac{2}{1}\right)^{1/2}

\left(\frac{3}{2}\right)^{1/4}

\left(\frac{4}{3}\right)^{1/8}

\left(\frac{5}{4}\right)^{1/16}

\dots</math>

which can then be compactly represented in infinite product form by:

:<math>\sigma = \prod_{k=1}^{\infty} \left(1+ \frac{1}{k}\right)^{1/2^k}</math>

Another product representation is given by:

:<math>\sigma = \prod_{n=1}^\infty\prod_{k=0}^n (k+1)^{(-1)^{k+n} \binom{n}{k</math>

Expressions for <math>\ln\sigma</math> include:

:<math>\ln \sigma = \sum_{k=1}^{\infty} \frac{\ln k}{2^k}</math>

:<math>\ln \sigma = \sum_{k=1}^{\infty} \frac{(-1)^{k+1{k} \text{Li}_k\left(\tfrac12\right)</math>

:<math>\ln \frac\sigma2 = \sum_{k=1}^{\infty} \frac{1}{2^k}\left(\ln\left(1+\frac{1}{k}\right)-\frac1k\right)</math>

Integrals

Integrals for <math>\ln\sigma</math> are given by:

:<math>\ln \sigma = \int_0^1 \frac{1-x}{(x-2)\ln x} dx</math>

:<math>\ln \sigma = \int_0^1 \int_0^1 \frac{-x}{(2-xy)\ln(xy)} dx dy</math>

Other formulas

The constant <math>\sigma</math> arises when studying the asymptotic behaviour of the sequence

:<math>g_0 = 1</math>

:<math>g_n = n g_{n-1}^2, \qquad n \ge 1</math>

with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows:

:<math>\gamma(\tfrac12)=2\ln\frac2 \sigma</math>

Universality

One may define a "continued binary expansion" for all real numbers in the set <math>(0,1]</math>, similarly to the decimal expansion or simple continued fraction expansion. This is done by considering the unique base-2 representation for a number <math>x\in(0,1]</math> which does not contain an infinite tail of 0's (for example write one half as <math>0.01111..._2</math> instead of <math>0.1_2</math>). Then define a sequence <math>(a_k)\sube \N</math> which gives the difference in positions of the 1's in this base-2 representation. This expansion for <math>x</math> is now given by:

<math>x=\langle a_1, a_2, a_3, ... \rangle</math>

thumb|The geometric means of the terms of [[Pi and e appear to tend to Somos' constant.|400x400px]]

For example the fractional part of Pi we have:

The first 1 occurs on position 3 after the radix point. The next 1 appears three places after the first one, the third 1 appears five places after the second one, etc. By continuing in this manner, we obtain:

<math>\pi-3= \langle 3, 3, 5, 1, 1, 1, 1 ... \rangle</math>

This gives a bijective map <math>(0,1] \mapsto \N ^\N </math>, such that for every real number <math>x\in(0,1]</math> we uniquely can give: