Somos' quadratic recurrence constant
Encyclopedia
In mathematics
, Somos' quadratic recurrence constant, named after Michael Somos
, is the number
This can be easily re-written into the far more quickly converging product representation
The constant σ arises when studying the asymptotic behaviour of the sequence
with first few terms 1, 1, 2, 12, 576, 1658880 ... . This sequence can be shown to have asymptotic behaviour as follows:
Guillera and Sondow give a representation in terms of the derivative
of the Lerch transcendent:
where ln is the natural logarithm
and Φ(z, s, q) is the Lerch transcendent.
Using series acceleration
it is the sum of the n-th differences of ln(k) at k=1 as given by:
Finally,
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Somos' quadratic recurrence constant, named after Michael Somos
Michael Somos
Michael Somos is an American mathematician, currently a visiting scholar in Georgetown University. In the late eighties he proposed an intriguing conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers...
, is the number
This can be easily re-written into the far more quickly converging product representation
The constant σ arises when studying the asymptotic behaviour of the sequence
with first few terms 1, 1, 2, 12, 576, 1658880 ... . This sequence can be shown to have asymptotic behaviour as follows:
Guillera and Sondow give a representation in terms of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the Lerch transcendent:
where ln is the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
and Φ(z, s, q) is the Lerch transcendent.
Using series acceleration
Series acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration...
it is the sum of the n-th differences of ln(k) at k=1 as given by:
Finally,
.