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 and also in connection to the binary representations of real numbers between zero and one. The constant named after Michael Somos. It is defined by:

:$\sigma = \sqrt$

which gives a numerical value of approximately:

:$\sigma = 1.661687949633594121295\dots\;$ .

Sums and products

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

:$\sigma=\prod_^\infty k^ = 1^\;2^\; 3^\; 4^ \dots$

This can be easily rewritten into the far more quickly converging product representation
:$\sigma = \left(\frac\right)^ \left(\frac\right)^ \left(\frac\right)^ \left(\frac\right)^ \dots$

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

:$\sigma = \prod_^ \left(1+ \frac\right)^$
Another product representation is given by:
:$\sigma = \prod_^\infty\prod_^n (k+1)^$
Expressions for $\ln\sigma$ include:

:$\ln \sigma = \sum_^ \frac$

:$\ln \sigma = \sum_^ \frac \text_k\left(\tfrac12\right)$

:$\ln \frac\sigma2 = \sum_^ \frac\left(\ln\left(1+\frac\right)-\frac1k\right)$

Integrals

Integrals for $\ln\sigma$ are given by:

:$\ln \sigma = \int_0^1 \frac dx$

:$\ln \sigma = \int_0^1 \int_0^1 \frac dx dy$

Other formulas

The constant $\sigma$ arises when studying the asymptotic behaviour of the sequence
:$g_0 = 1$
:$g_n = n g_^2, \qquad n \ge 1$

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

:$g_n \sim \left(n+2-n^+4n^-21n^+138n^+O(n^)\right)^$
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent $\Phi(z, s, q)$:
:$\ln\sigma = -\frac \frac\!\left( 1/2, 0, 1 \right)$
If one defines the Euler-constant function (which gives Euler's constant for $z=1$) as:

:$\gamma(z)=\sum_^\infty z^\left(\frac1n - \ln\left(\frac\right)\right)$

one has:

:$\gamma(\tfrac12)=2\ln\frac2 \sigma$

Universality

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

$x=\langle a_1, a_2, a_3, ... \rangle$

For example the fractional part of Pi we have:

$\ = 0.14159 \,26535 \, 89793... = 0.00100 \, 10000 \, 11111 ..._2$

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:

$\pi-3= \langle 3, 3, 5, 1, 1, 1, 1 ... \rangle$

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

$x = \langle a_1, a_2, a_3, ... \rangle :\Leftrightarrow x= \sum _^\infty 2^$

It can now be proven that for almost all numbers $x\in(0,1]$ the limit of the geometric mean of the terms $a_k$ converges to Somos' constant. That is, for almost all numbers in that interval we have:

\sigma = \lim_\sqrt /math>

Somos' constant is universal for the "continued binary expansion" of numbers $x\in(0,1]$ in the same sense that Khinchin's constant is universal for the simple continued fraction expansions of numbers $x\in\R$.

Generalizations

The ''generalized Somos' constants'' may be given by:
:$\sigma_t=\prod_^\infty k^ = 1^\;2^\; 3^\; 4^\dots$
for $t>1$.

The following series holds:
:$\ln\sigma_t=\sum_^\infty \frac$
We also have a connection to the Euler-constant function:
:$\gamma(\tfrac1t)=t\ln\left(\frac\right)$
and the following limit, where $\gamma$ is Euler's constant:
:$\lim_ t\sigma_^=e^$

See also

*Euler's constant
*Khinchin's constant
*Binary number
*Ergodic theory
*List of mathematical constants

References

{{reflist

Mathematical constants
Infinite products