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| In [[mathematics]], '''Somos' quadratic recurrence constant''', named after [[Michael Somos]], is the number
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| :<math>\sigma = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} =
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| 1^{1/2}\;2^{1/4}\; 3^{1/8} \cdots.\,</math>
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| This can be easily re-written into the far more quickly converging product representation
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| :<math>\sigma = \sigma^2/\sigma =
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| \left(\frac{2}{1} \right)^{1/2}
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| \left(\frac{3}{2} \right)^{1/4}
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| \left(\frac{4}{3} \right)^{1/8}
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| \left(\frac{5}{4} \right)^{1/16}
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| \cdots.</math>
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| The constant σ arises when studying the asymptotic behaviour of the sequence
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| :<math>g_0=1\, ; \, g_n = ng_{n-1}^2, \qquad n > 1, \, </math> | |
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| with first few terms 1, 1, 2, 12, 576, 1658880 ... {{OEIS|id=A052129}}. This sequence can be shown to have asymptotic behaviour as follows:<ref>{{MathWorld|title=Somos's Quadratic Recurrence Constant|urlname=SomossQuadraticRecurrenceConstant}}</ref>
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| :<math>g_n \sim \frac {\sigma^{2^n}}{n + 2 + O(\frac{1}{n})}. </math>
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| Guillera and Sondow give a representation in terms of the [[derivative]] of the [[Lerch transcendent]]:
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| :<math>\ln \sigma = \frac{-1}{2}
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| \frac {\partial \Phi} {\partial s}
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| \left( \frac{1}{2}, 0, 1 \right)</math>
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| where ln is the [[natural logarithm]] and <math>\Phi</math>(''z'', ''s'', ''q'') is the Lerch transcendent.
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| Using [[series acceleration]] it is the sum of the n-th differences of ln(k) at k=1 as given by:
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| :<math>\ln \sigma = \sum_{n=1}^\infty \sum_{k=0}^n (-1)^{n-k} {n \choose k} \ln (k+1). </math>
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| Finally,
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| :<math> \sigma = 1.661687949633594121296\dots\;</math> {{OEIS|id=A112302}}.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| * Steven R. Finch, ''Mathematical Constants'' (2003), [[Cambridge University Press]], p. 446. ISBN 0-521-81805-2.
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| * Jesus Guillera and [[Jonathan Sondow]], "Double integrals and infinite products for some classical constants via analytic continuations of [[Lerch's transcendent]]", ''Ramanujan Journal'' 16 (2008), 247–270 (Provides an integral and a series representation). {{arxiv|math/0506319}}
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| {{refend}}
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| [[Category:Mathematical constants]]
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