Somos' quadratic recurrence constant: Difference between revisions

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{{Short description|Mathematical constant}}

In [[mathematics]], '''Somo's quadratic recurrence constant''' is defined as the number

In [[mathematical analysis]] and [[number theory]], '''Somos' quadratic recurrence constant''' or simply '''Somos' constant''' is a [[Mathematical constant|constant]] defined as an expression of infinitely many nested [[Square root|square roots]]. It arises when studying the [[Asymptotic analysis|asymptotic behaviour]] of a certain [[sequence]]<ref name=":3" /> and also in connection to the [[Binary number|binary representations]] of [[Real number|real numbers]] between [[0|zero]] and [[1|one]].<ref name=":4">{{cite arXiv |last=Neunhäuserer |first=Jörg |title=On the universality of Somos' constant |date=2020-10-13 |class=math.DS |eprint=2006.02882}}</ref> The constant named after [[Michael Somos]]. It is defined by:

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

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

which gives a numerical value of approximately:<ref>{{Cite journal |last=Hirschhorn |first=Michael D. |date=2011-11-01 |title=A note on Somosʼ quadratic recurrence constant |url=https://www.sciencedirect.com/science/article/pii/S0022314X11001284 |journal=Journal of Number Theory |volume=131 |issue=11 |pages=2061–2063 |doi=10.1016/j.jnt.2011.04.010 |issn=0022-314X}}</ref>

Sondow gives a representation in terms of the [[Lerch zeta function]]

:<math>\sigma = 1.661687949633594121295\dots\;</math> {{OEIS|id=A112302}}.

:xxx

==Sums and products==

and the series representation

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

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

:xxx

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

This can be easily rewritten into the far more quickly [[limit (mathematics)|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:<ref name=":0">{{MathWorld|title=Somos's Quadratic Recurrence Constant|urlname=SomossQuadraticRecurrenceConstant}}</ref>

:<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> {{OEIS|id=A114124}} include:<ref name=":0" /><ref>{{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|doi-access=free }}</ref>

:<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:<ref name=":0" /><ref name=":1">{{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}}</ref>

:<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<ref name=":3">{{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}}</ref>

:<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, ... {{OEIS|id=A052129}}. This sequence can be shown to have asymptotic behaviour as follows:<ref name=":0" />

:<math>g_n \sim {\sigma^{2^n}}\left(n+2-n^{-1}+4n^{-2}-21n^{-3}+138n^{-4}+O(n^{-5})\right)^{-1} </math>

Guillera and Sondow give a representation in terms of the [[derivative]] of the [[Lerch transcendent]] <math>\Phi(z, s, q)</math>:<ref name=":1" />

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

If one defines the [[Euler's constant#Euler-constant function|Euler-constant function]] (which gives [[Euler's constant]] for <math>z=1</math>) as:

:<math>\gamma(z)=\sum_{n=1}^\infty z^{n-1}\left(\frac1n - \ln\left(\frac{n+1}{n}\right)\right)</math>

one has:<ref>{{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}}</ref><ref name=":2">{{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 }}</ref><ref>{{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|url-access=subscription }}</ref>

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

==Universality==

One may define a ''"continued binary expansion"'' for all real numbers in the [[Set (mathematics)|set]] <math>(0,1]</math>, similarly to the [[decimal expansion]] or [[Simple continued fraction|simple continued fraction expansion]]. This is done by considering the unique [[Base-2|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:<ref name=":5">{{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|url-access=subscription }}</ref>

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

[[File:SomosConstant.png|thumb|The geometric means of the terms of [[Pi]] and [[E (mathematical constant)|e]] appear to tend to Somos' constant.|400x400px]]

For example the [[fractional part]] of [[Pi]] we have:

<math>\{\pi\} = 0.14159 \,26535 \, 89793... = 0.00100 \, 10000 \, 11111 ..._2 </math> {{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:

<math>\pi-3= \langle 3, 3, 5, 1, 1, 1, 1 ... \rangle</math> {{OEIS|A320298}}

This gives a [[Bijection|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:<ref name=":5" />

<math>x = \langle a_1, a_2, a_3, ... \rangle :\Leftrightarrow x= \sum _{k=1}^\infty 2^{-(a_1+...+a_k)}</math>

It can now be proven that for almost all numbers <math>x\in(0,1]</math> the limit of the [[geometric mean]] of the terms <math>a_k</math> converges to Somos' constant. That is, for almost all numbers in that interval we have:<ref name=":4" />

<math>\sigma = \lim_{n\to\infty}\sqrt[n]{a_1a_2...a_n}</math>

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

==Generalizations==

The ''generalized Somos' constants'' may be given by:

:<math>\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</math>

for <math>t>1</math>.

The following series holds:

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

We also have a connection to the [[Euler's constant#Euler-constant function|Euler-constant function]]:<ref name=":2" />

:<math>\gamma(\tfrac1t)=t\ln\left(\frac{t}{(t-1)\sigma_t^{t-1}}\right)</math>

and the following limit, where <math>\gamma</math> is [[Euler's constant]]:

:<math>\lim_{t\to 0^+} t\sigma_{t+1}^{t}=e^{-\gamma}</math>

==See also==

*[[Euler's constant]]

*[[Khinchin's constant]]

*[[Binary number]]

*[[Ergodic theory]]

*[[List of mathematical constants]]

==References==

{{reflist}}

* S. Finch, ''Mathematical Constants'', (2003) Cambridge University Press, Cambridge p.446

* Jesus Guillera and Jonathan Sondow, ''[http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent]'' (2005) ''(Provides an integral and a series representation)''.

{{math-stub}}

[[Category:Mathematical constants]]

[[Category:Infinite products]]