Somos' quadratic recurrence constant explained

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.^[1] The constant named after Michael Somos. It is defined by:

\sigma=\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{5 … }}}}}

which gives a numerical value of approximately:^[2]

\sigma=1.661687949633594121295...

Sums and products

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

	infty
\sigma=\prod
	k=1

	1/2^k
k

=1^1/2 2^1/4 3^1/8 4^1/16...

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

\sigma=\left(

	2
	1

\right)^1/2\left(

	3
	2

\right)^1/4\left(

	4
	3

\right)^1/8\left(

	5
	4

\right)^1/16...

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

\sigma=

	infty
\prod
	k=1

\left(1+

	1
	k

	1/2^k
\right)

Another product representation is given by:

\sigma=

	n
\prod
	k=0

	(-1)^k+n\binom{n
(k+1)

{k}}

Expressions for

ln\sigma

include:^[3]

ln\sigma=

	infty
\sum
	k=1

	lnk
	2^k

ln\sigma=

	infty
\sum
	k=1

	(-1)^k+1
	k

Li_{k\left(\tfrac12\right)}

	\sigma2
	=

	infty
\sum
	k=1

	1	\left(ln\left(1+
	2^k

	1	\right)-
	k

	1k\right)

Integrals

Integrals for

ln\sigma

are given by:^[4]

ln\sigma=

	1
\int
	0

	1-x
	(x-2)lnx

ln\sigma=

	1
\int
	0

	1
\int
	0

	-x
	(2-xy)ln(xy)

dxdy

Other formulas

The constant

\sigma

arises when studying the asymptotic behaviour of the sequence^[5]

g₀=1

g_n=n

	2,
g
	n-1

n\ge1

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

g_n\sim

	2ⁿ
{\sigma

}\left(n+2-n^+4n^-21n^+138n^+O(n^)\right)^ Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent

\Phi(z,s,q)

ln\sigma=-

	1
	2

	\partial\Phi
	\partials

\left(1/2,0,1\right)

If one defines the Euler-constant function (which gives Euler's constant for

z=1

) as:

	infty
\gamma(z)=\sum
	n=1

z^n-1\left(

	1n	ln\left(
	-

	n+1
	n

\right)\right)

one has:^[6] ^[7] ^[8]

\gamma(\tfrac12)=2ln	2
	\sigma

Universality

(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

0.1₂

). Then define a sequence

(a_k)\sube\N

which gives the difference in positions of the 1's in this base-2 representation. This expansion for

is now given by:^[9]

x=\langlea_1,a_2,a_3,...\rangle

For example the fractional part of Pi we have:

\{\pi\}=0.141592653589793...=0.001001000011111...₂

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=\langle3,3,5,1,1,1,1...\rangle

This gives a bijective map

(0,1]\mapsto\N^\N

, such that for every real number

x\in(0,1]

we uniquely can give:

x=\langlea_1,a_2,a_3,...\rangle:\Leftrightarrowx=\sum

	infty

	k=1

	-(a_1+...+a_k)
2

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\toinfty\sqrt[n]{a_1a_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

	infty

	k=1

	1/t^k
k

=1^1/t

	1/t²
2

	1/t³
3

	1/t⁴
4

...

for

t>1

The following series holds:

ln\sigma_t=\sum

	infty

	k=1

	lnk
	t^k

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

\gamma(\tfrac1t)=tln\left(

	t-1
(t-1)\sigma
	t

\right)

and the following limit, where

\gamma

is Euler's constant:

\lim
	t\to0⁺

	t
t\sigma
	t+1

=e^-\gamma

Notes and References

Neunhäuserer . Jörg . On the universality of Somos' constant . 2020-10-13 . math.DS . 2006.02882.
Hirschhorn . Michael D. . 2011-11-01 . A note on Somosʼ quadratic recurrence constant . Journal of Number Theory . 131 . 11 . 2061–2063 . 10.1016/j.jnt.2011.04.010 . 0022-314X.
Mortici . Cristinel . 2010-12-01 . Estimating the Somos' quadratic recurrence constant . Journal of Number Theory . 130 . 12 . 2650–2657 . 10.1016/j.jnt.2010.06.012 . 0022-314X.
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 . 10.1007/s11139-007-9102-0 . math/0506319 . 1382-4090.
Book: Finch, Steven R. . Mathematical Constants . 2003-08-18 . Cambridge University Press . 978-0-521-81805-6 . en.
Chen . Chao-Ping . Han . Xue-Feng . 2016-09-01 . On Somos' quadratic recurrence constant . Journal of Number Theory . 166 . 31–40 . 10.1016/j.jnt.2016.02.018 . 0022-314X.
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 . 10.1016/j.jmaa.2006.09.081. math/0610499 . 2007JMAA..332..292S .
Pilehrood . Khodabakhsh Hessami . Pilehrood . Tatiana Hessami . 2007-01-01 . Arithmetical properties of some series with logarithmic coefficients . Mathematische Zeitschrift . en . 255 . 1 . 117–131 . 10.1007/s00209-006-0015-1 . 1432-1823.
Neunhäuserer . Jörg . 2011-11-01 . On the Hausdorff dimension of fractals given by certain expansions of real numbers . Archiv der Mathematik . en . 97 . 5 . 459–466 . 10.1007/s00013-011-0320-8 . 1420-8938.

Somos' quadratic recurrence constant explained

Sums and products

Integrals

Other formulas

Universality

Generalizations

See also

Notes and References