A constant which appears in estimates of combinatorial functions. It is usually denoted , , or . It is
not known if is Irrational. Numerically,

(1) |

(2) | |||

(3) | |||

(4) | |||

(5) | |||

(6) |

where is the Dirichlet Beta Function. In terms of the Polygamma Function ,

(7) | |||

(8) | |||

(9) |

Applying Convergence Improvement to (3) gives

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 807-808, 1972.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 551-552, 1985.

Fee, G. J. ``Computation of Catalan's Constant using Ramanujan's Formula.''
*ISAAC '90. Proc. Internat. Symp. Symbolic Algebraic Comp., Aug. 1990.*
Reading, MA: Addison-Wesley, 1990.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/catalan/catalan.html

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Glaisher, J. W. L. ``Numerical Values of the Series
for , 4, 6.''
*Messenger Math.* **42**, 35-58, 1913.

Gosper, R. W. ``A Calculus of Series Rearrangements.'' In *Algorithms and Complexity: New Directions and
Recent Results* (Ed. J. F. Traub). New York: Academic Press, 1976.

Nielsen, N. *Der Eulersche Dilogarithms.* Leipzig, Germany: Halle, pp. 105 and 151, 1909.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Sloane, N. J. A. Sequences
A014538 and
A006752/M4593
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Srivastava, H. M. and Miller, E. A. ``A Simple Reducible Case of Double Hypergeometric Series involving
Catalan's Constant and Riemann's Zeta Function.'' *Int. J. Math. Educ. Sci. Technol.* **21**, 375-377, 1990.

Vardi, I. *Computational Recreations in Mathematica.* Reading, MA: Addison-Wesley, p. 159, 1991.

Yang, S. ``Some Properties of Catalan's Constant .'' *Int. J. Math. Educ. Sci. Technol.* **23**, 549-556, 1992.

© 1996-9

1999-05-26