Somos' quadratic recurrence constant

In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number

$\sigma ={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots .\,$

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

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

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

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

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

$g_{0}=1\,;\,g_{n}=ng_{n-1}^{2},\qquad n>1,\,$

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:

$g_{n}\sim {\frac {\sigma ^{2^{n}}}{n+2+O(1/n)}}.$

Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent:

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

where $\ln$ is the natural logarithm and $\Phi (z,s,q)$ is the Lerch transcendent.

Finally,

$\sigma =1.661687949633594121296\dots \;$ (sequence A112302 in the OEIS).

References

• Steven R. Finch, Mathematical Constants (2003), Cambridge University Press, p. 446. ISBN 0-521-81805-2.
• Jesus Guillera and Jonathan Sondow, "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", Ramanujan Journal 16 (2008), 247–270 (Provides an integral and a series representation). arXiv:math/0506319