Somos' quadratic recurrence constant

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

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

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

[math]\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, }[/math]

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

[math]\displaystyle{ \sigma = \prod_{k=1}^{\infty} \left(1 + \frac{1}{k}\right)^{\frac{1}{2^k}}. }[/math]

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

[math]\displaystyle{ g_0 = 1\, ; \,g_n = n g_{n-1}^2, \qquad n \gt 1,\, }[/math]

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:

[math]\displaystyle{ g_n \sim \frac {\sigma^{2^n}}{n + 2 + O(\frac{1}{n})}. }[/math]

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

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

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

Finally,

[math]\displaystyle{ \sigma = 1.661687949633594121296\dots\; }[/math] (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