Somos' quadratic recurrence constant

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

${\displaystyle \sigma ={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots .\,}$

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

${\displaystyle \sigma =\sigma ^{2}/\sigma =\left({\frac {2}{1}}\right)^{1/2}\left({\frac {3}{2}}\right)^{1/4}\left({\frac {4}{3}}\right)^{1/8}\left({\frac {5}{4}}\right)^{1/16}\cdots ,}$

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

${\displaystyle \sigma =\prod _{k=1}^{\infty }\left(1+{\frac {1}{k}}\right)^{\frac {1}{2^{k}}}.}$

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

${\displaystyle g_{0}=1\,;\,g_{n}=ng_{n-1}^{2},\qquad n>1,\,}$

with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence A052129 in the OEIS). This sequence can be shown to have asymptotic behaviour as follows:^[1]

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

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

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

where ln is the natural logarithm and ${\displaystyle \Phi }$(z, s, q) is the Lerch transcendent.

Finally,

${\displaystyle \sigma =1.661687949633594121296\dots \;}$ (sequence A112302 in the OEIS).

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