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In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, who is a researcher in the Georgetown University Mathematics Department, is the number

$sigma = sqrt \left\{1 sqrt \left\{2 sqrt\left\{3 cdots\right\}\right\}\right\} =$
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.

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

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

where ln is the natural logarithm and Φ(zsq) is the Lerch transcendent.

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

$ln sigma=sum_\left\{n=1\right\}^infty \left(-1\right)^n$
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, (2005) (Provides an integral and a series representation).
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