In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by
where β is the Dirichlet beta function. Its numerical value  is approximately (sequence A006752 in OEIS)
It is not known whether G is rational or irrational.
Catalan's constant was named after Eugène Charles Catalan.
Some identities include
where K(x) is a complete elliptic integral of the first kind, and
G appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:
Simon Plouffe gives an infinite collection of identities between the trigamma function, π2 and Catalan's constant; these are expressible as paths on a graph.
It also appears in connection with the hyperbolic secant distribution.
Quickly converging series
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
The theoretical foundations for such series is given by Broadhurst.
The number of known digits of Catalan's constant G has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.
- Victor Adamchik, 33 representations for Catalan's constant (undated)
- Adamchik,, Victor (2002). "A certain series associated with Catalan's constant". Zeitschr. f. Analysis und ihre Anwendungen (ZAA) 21 (3): 1–10. MR1929434. http://www-2.cs.cmu.edu/~adamchik/articles/csum.html.
- Simon Plouffe, A few identities (III) with Catalan, (1993) (Provides over one hundred different identities).
- Simon Plouffe, A few identities with Catalan constant and Pi^2, (1999) (Provides a graphical interpretation of the relations)
- Weisstein, Eric W., "Catalan's Constant" from MathWorld.
- Catalan constant: Generalized power series at the Wolfram Functions Site
- Greg Fee, Catalan's Constant (Ramanujan's Formula) (1996) (Provides the first 300,000 digits of Catalan's constant.).
- Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal 3 (2): 159–173. doi:10.1023/A:1006945407723. MR1703281.
- Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". arΧiv:0706.0356.
- Bradley, David M. (2001), Representations of Catalan's constant
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