Somos' quadratic recurrence constant

In mathematics, Somos' quadratic recurrence constant is defined as the number

$\sigma ={\sqrt {1{\sqrt {2{\sqrt {3\ldots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\ldots$

This can be easily re-written into the far more quickly converging product representation

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

Sondow gives a representation in terms of the derivative of the Lerch transcendent:

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

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

A series representation, as a sum over the binomial coefficient, is also given:

$\ln \sigma =\sum _{n=1}^{\infty }(-1)^{n}\sum _{k=0}^{n}(-1)^{k}{n \choose k}\ln(k+1)$

Finally,

$\sigma =1.661687949335941212958...\;$

References

• S. Finch, Mathematical Constants, (2003) Cambridge University Press, Cambridge p.446
• Jesus Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent (2005) (Provides an integral and a series representation).