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:

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

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

$g_0=1;g_n=n{g}_{n-1}^2,n>1,$

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:

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

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

$\ln \sigma =\frac{-1}{2}\frac{\partial \mathrm{\Phi }}{\partial s}(\frac{1}{2},0,1)$

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

Finally,

$\sigma =1.661687949633594121296\dots$ (sequence A112302 in the OEIS).

References

• Steven R. Finch, Mathematical Constants (2003), Cambridge University Press , p. 446. ISBN:0-521-81805-2.
• 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