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.

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 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, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent (2005) (Provides an integral and a series representation).