Somos' quadratic recurrence constant について

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Somos' quadratic recurrence constant ：ウィキペディア英語版

Somos' quadratic recurrence constant
In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number
:

\sigma = \sqrt  = 1^\;2^\; 3^ \cdots.\,

This can be easily re-written into the far more quickly converging product representation
:

\sigma = \sigma^2/\sigma = \left(\frac \right)^\left(\frac \right)^\left(\frac \right)^\left(\frac \right)^\cdots.

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

g_0=1\, ; \, g_n = ng_^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 )}.

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

\ln \sigma = \frac \frac   \left( \frac, 0, 1 \right)

where ln is the natural logarithm and

\Phi

(''z'', ''s'', ''q'') is the Lerch transcendent.
Using series acceleration it is the sum of the n-th differences of ln(k) at k=1 as given by:
:

\ln \sigma = \sum_^\infty \sum_^n (-1)^  \ln (k+1).

Finally,
:

\sigma = 1.661687949633594121296\dots\;

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