Somos' quadratic recurrence constant

Template:Short description 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:

$\sigma =1.661687949633594121295\dots$ Template:OEIS.

Sums and products

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

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

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:

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

Another product representation is given by:

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

Expressions for $\ln \sigma$ Template:OEIS include:

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

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

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

Integrals

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

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

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

Other formulas

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

$g_0=1$

$g_n=n{g}_{n-1}^2,n\ge 1$

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

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

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

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

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

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

one has:

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

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:

$x=⟨a_1,a_2,a_3,...⟩$

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:

$\{\pi \}=0.141592653589793...=0.001001000011111..{.}_2$ Template: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:

$\pi -3=⟨3,3,5,1,1,1,1...⟩$ Template: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:

$x=⟨a_1,a_2,a_3,...⟩:\iff x=\sum _{k=1}^{\mathrm{\infty }}{2}^{-(a_1+...+a_k)}$

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:

$\sigma =\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{a_1{a}_2...a_n}$

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 $x\in ℝ$.

Generalizations

The generalized Somos' constants may be given by:

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

for $t>1$.

The following series holds:

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

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

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

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

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

See also

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