Somos' quadratic recurrence constant

In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, who is a researcher in the Georgetown University Mathematics Department, 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 re-written into the far more quickly converging product representation

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

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

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

where ln is the natural logarithm and Φ(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,

$\sigma =1.661687949633594121296\dots \;$

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,Ramanujan Journal 16 (2008), 247-270 (Provides an integral and a series representation).