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

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

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

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

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,

:$sigma\; =\; 1.661687949633594121296dots;$

References

* S. Finch, "Mathematical Constants", (2003) Cambridge University Press, Cambridge p.446
* Jesus Guillera and Jonathan Sondow, " [http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent] " (2005) "(Provides an integral and a series representation)".