Somos' quadratic recurrence constant

In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, 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,$$

which can then be compactly represented in infinite product form by:


 * $$\sigma = \prod_{k=1}^{\infty} \left(1 + \frac{1}{k}\right)^{\frac{1}{2^k}}.$$

The constant σ arises when studying the asymptotic behaviour of the sequence


 * $$g_0 = 1\, ; \,g_n = n g_{n-1}^2, \qquad n > 1,\,$$

with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows:


 * $$g_n \sim \frac {\sigma^{2^n}}{n + 2 + O(\frac{1}{n})}. $$

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 $$\Phi$$(z, s, q) is the Lerch transcendent.

Finally,


 * $$\sigma = 1.661687949633594121296\dots\;$$.