Somos' quadratic recurrence constant: Difference between revisions

Revision as of 04:08, 13 July 2005

Revision as of 04:17, 13 July 2005

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1^{1/2}\;2^{1/4}\; 3^{1/8} \ldots</math>

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

Sondow gives a representation in terms of the [[Lerch zeta function]]

:<math>\sigma = \sigma^2/\sigma =

:xxx

\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}

\ldots</math>

Sondow gives a representation in terms of the derivative of the [[Lerch zeta functiontranscendant]]:

and the series representation

:<math>\ln \sigma = \frac{-1}{2}

:xxx

\frac {\partial \Phi} {\partial s}

\left( \frac{1}{2}, 0, 1 \right)</math>

where <math>\ln</math> is the [[natural logarithm]] and <math>\Phi(z,s,q)</math> is the Lerch transcendant.

and theA series representation is also given:

:<math>\ln \sigma=\sum_{n=1}^\infty </math>

==References==