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 ^[1] and also in connection to the binary representations of real numbers between zero and one. ^[2] The constant named after Michael Somos. It is defined by:

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

which gives a numerical value of approximately: ^[3]

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

Sums and products

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

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

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

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

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

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

Another product representation is given by: ^[4]

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

Expressions for ${\displaystyle \ln \sigma }$(sequence A114124 in the OEIS ) include: ^{[4] [5]}

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

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

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

Integrals

Integrals for ${\displaystyle \ln \sigma }$ are given by: ^{[4] [6]}

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

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

Other formulas

The constant ${\displaystyle \sigma }$ arises when studying the asymptotic behaviour of the sequence ^[1]

${\displaystyle g_{0}=1}$

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

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: ^[4]

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

Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent ${\displaystyle \Phi (z,s,q)}$: ^[6]

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

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

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

one has: ^{[7] [8] [9]}

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

Universality

One may define a "continued binary expansion" for all real numbers in the set ${\displaystyle (0,1]}$, similarly to the decimal expansion or simple continued fraction expansion. This is done by considering the unique base-2 representation for a number ${\displaystyle x\in (0,1]}$ which does not contain an infinite tail of 0's (for example write one half as ${\displaystyle 0.01111..._{2}}$ instead of ${\displaystyle 0.1_{2}}$). Then define a sequence ${\displaystyle (a_{k})\subseteq \mathbb {N} }$ which gives the difference in positions of the 1's in this base-2 representation. This expansion for ${\displaystyle x}$ is now given by: ^[10]

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

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:

${\displaystyle \{\pi \}=0.14159\,26535\,89793...=0.00100\,10000\,11111..._{2}}$(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:

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

This gives a bijective map ${\displaystyle (0,1]\mapsto \mathbb {N} ^{\mathbb {N} }}$, such that for every real number ${\displaystyle x\in (0,1]}$ we uniquely can give: ^[10]

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

It can now be proven that for almost all numbers ${\displaystyle x\in (0,1]}$ the limit of the geometric mean of the terms ${\displaystyle a_{k}}$ converges to Somos' constant. That is, for almost all numbers in that interval we have: ^[2]

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

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

Generalizations

The generalized Somos' constants may be given by:

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

for ${\displaystyle t>1}$.

The following series holds:

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

We also have a connection to the Euler-constant function: ^[8]

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

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

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

See also

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

References

1. Finch, Steven R. (2003-08-18). Mathematical Constants. Cambridge University Press. ISBN   978-0-521-81805-6.
2. Neunhäuserer, Jörg (2020-10-13), On the universality of Somos' constant, doi:10.48550/arXiv.2006.02882 , retrieved 2024-10-13
3. Hirschhorn, Michael D. (2011-11-01). "A note on Somosʼ quadratic recurrence constant". Journal of Number Theory. 131 (11): 2061–2063. doi:10.1016/j.jnt.2011.04.010. ISSN   0022-314X.
4. Weisstein, Eric W. "Somos's Quadratic Recurrence Constant". MathWorld .
5. Mortici, Cristinel (2010-12-01). "Estimating the Somos' quadratic recurrence constant". Journal of Number Theory. 130 (12): 2650–2657. doi:10.1016/j.jnt.2010.06.012. ISSN   0022-314X.
6. Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. doi:10.1007/s11139-007-9102-0. ISSN   1382-4090.
7. Chen, Chao-Ping; Han, Xue-Feng (2016-09-01). "On Somos' quadratic recurrence constant". Journal of Number Theory. 166: 31–40. doi:10.1016/j.jnt.2016.02.018. ISSN   0022-314X.
8. Sondow, Jonathan; Hadjicostas, Petros (2007). "The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant". Journal of Mathematical Analysis and Applications. 332 (1): 292–314. doi:10.1016/j.jmaa.2006.09.081.
9. Pilehrood, Khodabakhsh Hessami; Pilehrood, Tatiana Hessami (2007-01-01). "Arithmetical properties of some series with logarithmic coefficients". Mathematische Zeitschrift. 255 (1): 117–131. doi:10.1007/s00209-006-0015-1. ISSN   1432-1823.
10. Neunhäuserer, Jörg (2011-11-01). "On the Hausdorff dimension of fractals given by certain expansions of real numbers". Archiv der Mathematik. 97 (5): 459–466. doi:10.1007/s00013-011-0320-8. ISSN   1420-8938.