Somos' quadratic recurrence constant

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

:$\backslash sigma=\backslash prod\_\{k=1\}^\backslash infty\; k^\{1/2^k\}\; =$

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

:$\backslash sigma\; =$

\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:

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

Another product representation is given by:{{MathWorld|title=Somos's Quadratic Recurrence Constant|urlname=SomossQuadraticRecurrenceConstant}}

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

Expressions for $\backslash ln\backslash sigma$ {{OEIS|id=A114124}} include:{{Cite journal |last=Mortici |first=Cristinel |date=2010-12-01 |title=Estimating the Somos' quadratic recurrence constant |url=https://www.sciencedirect.com/science/article/pii/S0022314X10001897 |journal=Journal of Number Theory |volume=130 |issue=12 |pages=2650–2657 |doi=10.1016/j.jnt.2010.06.012 |issn=0022-314X}}

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

:$\backslash ln\; \backslash sigma\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{(-1)^\{k+1\}\}\{k\}\; \backslash text\{Li\}\_k\backslash left(\backslash tfrac12\backslash right)$

:$\backslash ln\; \backslash frac\backslash sigma2\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2^k\}\backslash left(\backslash ln\backslash left(1+\backslash frac\{1\}\{k\}\backslash right)-\backslash frac1k\backslash right)$

Integrals for $\backslash ln\backslash sigma$ are given by:{{Cite journal |last1=Guillera |first1=Jesus |last2=Sondow |first2=Jonathan |date=2008 |title=Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent |journal=The Ramanujan Journal |volume=16 |issue=3 |pages=247–270 |doi=10.1007/s11139-007-9102-0 |arxiv=math/0506319 |issn=1382-4090}}

:$\backslash ln\; \backslash sigma\; =\; \backslash int\_0^1\; \backslash frac\{1-x\}\{(x-2)\backslash ln\; x\}\; dx$

:$\backslash ln\; \backslash sigma\; =\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\{-x\}\{(2-xy)\backslash ln(xy)\}\; dx\; dy$

The constant $\backslash sigma$ arises when studying the asymptotic behaviour of the sequence{{Cite book |last=Finch |first=Steven R. |url=https://books.google.com/books?id=Pl5I2ZSI6uAC |title=Mathematical Constants |date=2003-08-18 |publisher=Cambridge University Press |isbn=978-0-521-81805-6 |language=en}}

:$g\_0\; =\; 1$

:$g\_n\; =\; n\; g\_\{n-1\}^2,\; \backslash qquad\; n\; \backslash ge\; 1$

with first few terms 1, 1, 2, 12, 576, 1658880, ... {{OEIS|id=A052129}}. This sequence can be shown to have asymptotic behaviour as follows:

:$g\_n\; \backslash sim\; \{\backslash sigma^\{2^n\}\}\backslash left(n+2-n^\{-1\}+4n^\{-2\}-21n^\{-3\}+138n^\{-4\}+O(n^\{-5\})\backslash right)^\{-1\}$

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

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

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

:$\backslash gamma(z)=\backslash sum\_\{n=1\}^\backslash infty\; z^\{n-1\}\backslash left(\backslash frac1n\; -\; \backslash ln\backslash left(\backslash frac\{n+1\}\{n\}\backslash right)\backslash right)$

one has:{{Cite journal |last1=Chen |first1=Chao-Ping |last2=Han |first2=Xue-Feng |date=2016-09-01 |title=On Somos' quadratic recurrence constant |url=https://www.sciencedirect.com/science/article/pii/S0022314X16300257 |journal=Journal of Number Theory |volume=166 |pages=31–40 |doi=10.1016/j.jnt.2016.02.018 |issn=0022-314X}}{{Cite journal |last1=Sondow |first1=Jonathan |last2=Hadjicostas |first2=Petros |date=2007 |title=The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant |journal=Journal of Mathematical Analysis and Applications |volume=332 |issue=1 |pages=292–314 |doi=10.1016/j.jmaa.2006.09.081|arxiv=math/0610499 |bibcode=2007JMAA..332..292S }}{{Cite journal |last1=Pilehrood |first1=Khodabakhsh Hessami |last2=Pilehrood |first2=Tatiana Hessami |date=2007-01-01 |title=Arithmetical properties of some series with logarithmic coefficients |url=https://link.springer.com/article/10.1007/s00209-006-0015-1 |journal=Mathematische Zeitschrift |language=en |volume=255 |issue=1 |pages=117–131 |doi=10.1007/s00209-006-0015-1 |issn=1432-1823}}

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

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 $x\backslash in(0,1]$ 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 $(a\_k)\backslash sube\; \backslash N$ which gives the difference in positions of the 1's in this base-2 representation. This expansion for $x$ is now given by:{{Cite journal |last=Neunhäuserer |first=Jörg |date=2011-11-01 |title=On the Hausdorff dimension of fractals given by certain expansions of real numbers |url=https://link.springer.com/article/10.1007/s00013-011-0320-8 |journal=Archiv der Mathematik |language=en |volume=97 |issue=5 |pages=459–466 |doi=10.1007/s00013-011-0320-8 |issn=1420-8938}}

$x=\backslash langle\; a\_1,\; a\_2,\; a\_3,\; ...\; \backslash rangle$

File:SomosConstant.png and e appear to tend to Somos' constant.|400x400px]]

For example the fractional part of Pi we have:

$\backslash \{\backslash pi\backslash \}\; =\; 0.14159\; \backslash ,26535\; \backslash ,\; 89793...\; =\; 0.00100\; \backslash ,\; 10000\; \backslash ,\; 11111\; ...\_2$ {{OEIS|A004601}}

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:

$\backslash pi-3=\; \backslash langle\; 3,\; 3,\; 5,\; 1,\; 1,\; 1,\; 1\; ...\; \backslash rangle$ {{OEIS|A320298}}

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

$x\; =\; \backslash langle\; a\_1,\; a\_2,\; a\_3,\; ...\; \backslash rangle\; :\backslash Leftrightarrow\; x=\; \backslash sum\; \_\{k=1\}^\backslash infty\; 2^\{-(a\_1+...+a\_k)\}$

It can now be proven that for almost all numbers $x\backslash in(0,1]$ 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:

$\backslash sigma\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash sqrt[n]\{a\_1a\_2...a\_n\}$

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

The generalized Somos' constants may be given by:

:$\backslash sigma\_t=\backslash prod\_\{k=1\}^\backslash infty\; k^\{1/t^k\}\; =$

1^{1/t}\;2^{1/t^2}\; 3^{1/t^3}\; 4^{1/t^4}\dots

for $t>1$.

The following series holds:

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

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

:$\backslash gamma(\backslash tfrac1t)=t\backslash ln\backslash left(\backslash frac\{t\}\{(t-1)\backslash sigma\_t^\{t-1\}\}\backslash right)$

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

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

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

{{reflist}}

Category:Mathematical constants

Category:Infinite products