mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Somos' quadratic recurrence constant, named after

Michael Somos Michael Somos is an American mathematician, who was a visiting scholar in the Georgetown University Mathematics and Statistics department for four years and is a visiting scholar at Catholic University of America. In the late eighties he proposed a ...

, is the number :

\sigma = \sqrt  = 
1^\;2^\; 3^ \cdots.\,

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

\sigma = \sigma^2/\sigma = 
\left(\frac \right)^
\left(\frac \right)^
\left(\frac \right)^
\left(\frac \right)^
\cdots,

which can then be compactly represented in

infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...

form by: :

\sigma = \prod_^ \left(1 + \frac\right)^.

The constant σ arises when studying the asymptotic behaviour of the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

g_0 = 1\, ; \,g_n = n g_^2, \qquad n > 1,\,

with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows: :

g_n \sim \frac .

Guillera and Sondow give a representation in terms of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of the

Lerch transcendent In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who publis ...

: :

\ln \sigma = \frac \frac\!\left( \frac, 0, 1 \right)

where ln is the

natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...

and

\Phi

(''z'', ''s'', ''q'') is the Lerch transcendent. Finally, :

\sigma = 1.661687949633594121296\dots\;

Notes

References

* Steven R. Finch, ''Mathematical Constants'' (2003),

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, p. 446. . * 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). Mathematical constants Infinite products {{Math-stub