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:

$\sigma ={\sqrt {1{\sqrt {2{\sqrt {3{\sqrt {4{\sqrt {5\cdots }}}}}}}}}}$

which gives a numerical value of approximately:

$\sigma =1.661687949633594121295\dots \;$ (sequence A112302 in the OEIS).

Sums and products

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

$\sigma =\prod _{k=1}^{\infty }k^{1/2^{k}}=1^{1/2}\;2^{1/4}\;3^{1/8}\;4^{1/16}\dots$

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

$\sigma =\left({\frac {2}{1}}\right)^{1/2}\left({\frac {3}{2}}\right)^{1/4}\left({\frac {4}{3}}\right)^{1/8}\left({\frac {5}{4}}\right)^{1/16}\dots$

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

$\sigma =\prod _{k=1}^{\infty }\left(1+{\frac {1}{k}}\right)^{1/2^{k}}$

Another product representation is given by:

$\sigma =\prod _{n=1}^{\infty }\prod _{k=0}^{n}(k+1)^{(-1)^{k+n}{\binom {n}{k}}}$

Expressions for $\ln \sigma$ (sequence A114124 in the OEIS) include:

$\ln \sigma =\sum _{k=1}^{\infty }{\frac {\ln k}{2^{k}}}$

$\ln \sigma =\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}{\text{Li}}_{k}\left({\tfrac {1}{2}}\right)$

$\ln {\frac {\sigma }{2}}=\sum _{k=1}^{\infty }{\frac {1}{2^{k}}}\left(\ln \left(1+{\frac {1}{k}}\right)-{\frac {1}{k}}\right)$

Integrals

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

$\ln \sigma =\int _{0}^{1}{\frac {1-x}{(x-2)\ln x}}dx$

$\ln \sigma =\int _{0}^{1}\int _{0}^{1}{\frac {-x}{(2-xy)\ln(xy)}}dxdy$

Other formulas

The constant $\sigma$ arises when studying the asymptotic behaviour of the sequence

$g_{0}=1$

$g_{n}=ng_{n-1}^{2},\qquad n\geq 1$

with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence A052129 in the OEIS). This sequence can be shown to have asymptotic behaviour as follows:

$g_{n}\sim {\sigma ^{2^{n}}}\left(n+2-n^{-1}+4n^{-2}-21n^{-3}+138n^{-4}+O(n^{-5})\right)^{-1}$

Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent $\Phi (z,s,q)$:

$\ln \sigma =-{\frac {1}{2}}{\frac {\partial \Phi }{\partial s}}\!\left(1/2,0,1\right)$

If one defines the Euler-constant function (which gives Euler's constant for $z=1$) as:

$\gamma (z)=\sum _{n=1}^{\infty }z^{n-1}\left({\frac {1}{n}}-\ln \left({\frac {n+1}{n}}\right)\right)$

one has:

$\gamma ({\tfrac {1}{2}})=2\ln {\frac {2}{\sigma }}$

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 $(a_{k})\subseteq \mathbb {N}$ which gives the difference in positions of the 1's in this base-2 representation. This expansion for $x$ is now given by:

$x=\langle a_{1},a_{2},a_{3},...\rangle$

The geometric means of the terms of Pi and e appear to tend to Somos' constant.

For example the fractional part of Pi we have:

$\{\pi \}=0.14159\,26535\,89793...=0.00100\,10000\,11111..._{2}$ (sequence A004601 in the OEIS)

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:

$\pi -3=\langle 3,3,5,1,1,1,1...\rangle$ (sequence A320298 in the OEIS)

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

$x=\langle a_{1},a_{2},a_{3},...\rangle :\Leftrightarrow x=\sum _{k=1}^{\infty }2^{-(a_{1}+...+a_{k})}$

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:

$\sigma =\lim _{n\to \infty }{\sqrt[{n}]{a_{1}a_{2}...a_{n}}}$

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 $x\in \mathbb {R}$.

Generalizations

The generalized Somos' constants may be given by:

$\sigma _{t}=\prod _{k=1}^{\infty }k^{1/t^{k}}=1^{1/t}\;2^{1/t^{2}}\;3^{1/t^{3}}\;4^{1/t^{4}}\dots$

for $t>1$.

The following series holds:

$\ln \sigma _{t}=\sum _{k=1}^{\infty }{\frac {\ln k}{t^{k}}}$

We also have a connection to the Euler-constant function:

$\gamma ({\tfrac {1}{t}})=t\ln \left({\frac {t}{(t-1)\sigma _{t}^{t-1}}}\right)$

and the following limit, where $\gamma$ is Euler's constant:

$\lim _{t\to 0^{+}}t\sigma _{t+1}^{t}=e^{-\gamma }$

See also

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