Catalan's Constant

A constant which appears in estimates of combinatorial functions. It is usually denoted $K$, $\beta(2)$, or $G$. It is not known if $K$ is Irrational. Numerically,

$$K= 0.915 965 594 177\ldots$$ (1)

(Sloane's A006752). The Continued Fraction for $K$ is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (Sloane's A014538). $K$ can be given analytically by the following expressions,

$$K \equiv \beta(2)$$ (2)

$$= \sum_{k=0}^\infty {(-1)^k\over(2k+1)^2} = {1\over 1^2}-{1\over 3^2}+{1\over 5^2}+\ldots$$ (3)

$$= 1 + \sum_{n=1}^\infty {1\over(4n+1)^2}-{1\over 9}-\sum_{n=1}^\infty{1\over (4n+3)^2}$$ (4)

$$= \int_0^1 {\tan^{-1}x\,dx\over x}$$ (5)

$$= -\int_0^1 {\ln x\, dx\over 1+x^2},$$ (6)

where $\beta(z)$ is the Dirichlet Beta Function. In terms of the Polygamma Function $\Psi_1(x)$,

$$K = {\textstyle{1\over 16}}\Psi_1\left({{\textstyle{1\over 4}}}\right)- {\textstyle{1\over 16}} \Psi_1\left({{\textstyle{3\over 4}}}\right)$$ (7)

$$= {\textstyle{1\over 80}}\Psi_1({\textstyle{5\over 12}})+{\textstyle{1\over 80}}\Psi_1({\textstyle{1\over 12}})-{\textstyle{1\over 10}}\pi^2$$ (8)

$$= {\textstyle{1\over 32}}\Psi_1({\textstyle{1\over 8}})-{\textstyle{1\over 32}}\Psi_1({\textstyle{3\over 8}})-{\textstyle{1\over 16}}\sqrt{2}.$$ (9)

Applying Convergence Improvement to (3) gives

$$K={1\over 16} \sum_{m=1}^\infty (m+1){3^m-1\over 4^m}\zeta(m+2),$$ (10)

where $\zeta(z)$ is the Riemann Zeta Function and the identity

$${1\over(1-3z)^2}-{1\over(1-z)^2}=\sum_{m=1}^\infty (m+1){3^m-1\over 4^m} z^m$$ (11)

has been used (Flajolet and Vardi 1996). The Flajolet and Vardi algorithm also gives

$$K={1\over\sqrt{2}}\prod_{k=1}^\infty \left[{\left({1-{1\over... ...^k}}}\right){\zeta(2^k)\over\beta(2^k)}}\right]^{1/(2^{k+1})},$$ (12)

where $\beta(z)$ is the Dirichlet Beta Function. Glaisher (1913) gave

$$K=1-\sum_{n=1}^\infty {n\zeta(2n+1)\over 16^n}$$ (13)

(Vardi 1991, p. 159). W. Gosper used the related Formula

$$K={1\over\sqrt{2}} \left[{1\over\Psi(2)-1}\right]^{2^{1/2}}\... ...{k=2}^\infty \left[{1\over -\Psi(2^k)-1}\right]^{1/(2^{k+1})},$$ (14)

where

$$\Psi(m)={m\psi_{m-1}({\textstyle{1\over 4}})\over \pi^m(2^m-1)4^{m-1}B_m},$$ (15)

where $B_n$ is a Bernoulli Number and $\psi(x)$ is a Polygamma Function (Finch). The Catalan constant may also be defined by

$$K\equiv {\textstyle{1\over 2}}\int_0^1 K(k)\,dk,$$ (16)

where $K(k)$ (not to be confused with Catalan's constant itself, denoted $K$) is a complete Elliptic Integral of the First Kind.

$$K={\pi\ln 2\over 8}+\sum_{i=1}^\infty {a_i\over 2^{\left\lfloor{(i+1)/2}\right\rfloor }i^2},$$ (17)

where

$$\{a_i\}=\{\overline{1, 1, 1, 0, -1, -1, -1, 0}\}$$ (18)

is given by the periodic sequence obtained by appending copies of $\{1, 1, 1, 0, -1, -1, -1, 0\}$ (in other words, $a_i\equiv a_{[(i-1) {\rm\ (mod\ 8})]+1}$ for $i>8$) and $\left\lfloor{x}\right\rfloor$ is the Floor Function (Nielsen 1909).

See also Dirichlet Beta Function

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 $1-1/3^n+1/5^n-1/7^n+1/9^n-\&c$ for $n=2$, 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 $G$.'' Int. J. Math. Educ. Sci. Technol. 23, 549-556, 1992.