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Catalan's Constant

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Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted K (this work), G (e.g., Borwein et al. 2004, p. 49), or C (Wolfram Language).

Catalan's constant may be defined by

K=sum_(k=0)^infty((-1)^k)/((2k+1)^2)
(1)

(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if K is irrational.

Catalan's constant is implemented in the Wolfram Language as Catalan.

The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,

K=0.915965594177...
(2)

(OEIS A006752).

K can be given analytically by the following expressions

K=beta(2)
(3)
=-ichi_2(i)
(4)
=1/(24)pi-1/2pilnA+4pizeta^'(-1,1/4),
(5)

where beta(z) is the Dirichlet beta function, chi_nu(z) is Legendre's chi-function, A is the Glaisher-Kinkelin constant, and zeta^'(s,a) is the partial derivative of the Hurwitz zeta function with respect to the first argument.

Glaisher (1913) gave

K=1-sum_(n=1)^infty(nzeta(2n+1))/(16^n)
(6)

(Vardi 1991, p. 159). It is also given by the sums

K=sum_(k=0)^(infty)1/((4k+1)^2)-sum_(k=0)^(infty)1/((4k+3)^2)
(7)
=-1/8pi^2+2sum_(k=0)^(infty)1/((4k+1)^2)
(8)
=1/8pi^2-2sum_(k=0)^(infty)1/((4k+3)^2)
(9)

Equations (◇) and (◇) follow from

zeta(2)=sum_(n=1)^infty1/(n^2)=1/6pi^2,
(10)

together with

sum_(n=1,3,...)1/(n^2)=sum_(n=1)^(infty)1/(n^2)-sum_(n=2,4,...)^(infty)1/(n^2)
(11)
=zeta(2)-1/4sum_(n=1)^(infty)1/(n^2)
(12)
=3/4zeta(2)
(13)
=1/8pi^2.
(14)

But

sum_(n=1,3,...)1/(n^2)=sum_(k=0)^(infty)1/((4k+1)^2)+sum_(k=0)^(infty)1/((4k+3)^2)
(15)
=1/8pi^2,
(16)

so combining (16) with (◇) gives (◇) and (◇).

Applying convergence improvement to (◇) gives

K=1/(16)sum_(m=1)^infty(m+1)(3^m-1)/(4^m)zeta(m+2),
(17)

where zeta(z) is the Riemann zeta function and the identity

1/((1-3z)^2)-1/((1-z)^2)=sum_(m=1)^infty(m+1)(3^m-1)/(4^m)z^m
(18)

has been used (Flajolet and Vardi 1996).

A beautiful double series due to O. Oloa (pers. comm., Dec. 30, 2005) is given by

sum_(i=1)^inftysum_(j=1)^infty((i-1)!(j-1)!)/((i+j)!)(4^(i+j))/((2i+2j+1)(2(i+j); i+j))=8(1-K).
(19)

There are a large number of BBP-type formulas with coefficient (-1)^k, the first few being

K=sum_(k=0)^(infty)((-1)^k)/((2k+1)^2)
(20)
=4sum_(k=0)^(infty)((-1)^k)/((4k+2)^2)
(21)
=sum_(k=0)^(infty)(-1)^k[1/((6k+1)^2)-1/((6k+3)^2)+1/((6k+5)^2)]
(22)
=1/3sum_(k=0)^(infty)(-1)^k[2/((6k+1)^2)+7/((6k+3)^2)+2/((6k+5)^2)]
(23)
=sum_(k=0)^(infty)(-1)^k[1/((10k+1)^2)-1/((10k+3)^2)+1/((10k+5)^2)-1/((10k+7)^2)+1/((10k+9)^2)]
(24)
=1/3sum_(k=0)^(infty)(-1)^k[4/((10k+1)^2)-4/((10k+3)^2)-(21)/((10k+5)^2)-4/((10k+7)^2)+4/((10k+9)^2)]
(25)
=sum_(k=0)^(infty)(-1)^k[1/((14k+1)^2)-1/((14k+3)^2)+1/((14k+5)^2)-1/((14k+7)^2)+1/((14k+9)^2)-1/((14k+11)^2)+1/((14k+13)^2)]
(26)
=1/6sum_(k=0)^(infty)(-1)^k[5/((14k+1)^2)-5/((14k+3)^2)+5/((14k+5)^2)+(44)/((14k+7)^2)+5/((14k+9)^2)-5/((14k+11)^2)+5/((14k+13)^2)]
(27)

(E. W. Weisstein, Feb. 26, 2006).

BBP-type formula identities for K with higher powers include

K=3/(64)sum_(k=0)^(infty)((-1)^k)/(64^k)[(32)/((12k+1)^2)-(32)/((12k+2)^2)-(32)/((12k+3)^2)-8/((12k+5)^2)-(16)/((12k+6)^2)-4/((12k+7)^2)-4/((12k+9)^2)-2/((12k+10)^2)+1/((12k+11)^2)]
(28)

(V. Adamchik, pers. comm., Sep. 28, 2007),

K=5/(1024)sum_(k=0)^(infty)((-1)^k)/(1024^k)[(512)/((20k+1)^2)-(1536)/((20k+2)^2)+(256)/((20k+3)^2)+(512)/((20k+5)^2)+(384)/((20k+6)^2)-(64)/((20k+7)^2)+(32)/((20k+9)^2)+(64)/((20k+10)^2)+(64)/((20k+11)^2)+(16)/((20k+12)^2)-8/((20k+13)^2)+(24)/((20k+14)^2)+(16)/((20k+15)^2)+2/((20k+16)^2)+2/((20k+17)^2)-6/((20k+18)^2)+1/((20k+19)^2)]
(29)

(E. W. Weisstein, Sep. 30, 2007),

K=1/(1024)sum_(k=0)^(infty)1/(4096^k)[(3072)/((24k+1)^2)-(3072)/((24k+2)^2)-(23040)/((24k+3)^2)+(12288)/((24k+4)^2)-(768)/((24k+5)^2)+(9216)/((24k+6)^2)+(10368)/((24k+8)^2)+(2496)/((24k+9)^2)-(192)/((24k+10)^2)+(768)/((24k+12)^2)-(48)/((24k+13)^2)+(360)/((24k+15)^2)+(648)/((24k+16)^2)+(12)/((24k+17)^2)+(168)/((24k+18)^2)+(48)/((24k+20)^2)-(39)/((24k+21)^2)]
(30)

(Borwein and Bailey 2003, p. 128), and

K=1/(1024)sum_(k=0)^(infty)1/(4096^k)[(1024)/((24k+1)^2)+(1024)/((24k+2)^2)-(512)/((24k+3)^2)-(3072)/((24k+4)^2)-(256)/((24k+5)^2)-(2048)/((24k+6)^2)-(256)/((24k+7)^2)-(1152)/((24k+8)^2)-(320)/((24k+9)^2)+(64)/((24k+10)^2)+(64)/((24k+11)^2)-(16)/((24k+13)^2)+(64)/((24k+14)^2)+8/((24k+15)^2)-(72)/((24k+16)^2)+4/((24k+17)^2)-8/((24k+18)^2)+4/((24k+19)^2)-(12)/((24k+20)^2)+5/((24k+21)^2)+4/((24k+22)^2)-1/((24k+23)^2)]
(31)
=1/(3072)sum_(k=0)^(infty)1/(4096^k)[(5120)/((24k+1)^2)-(8192)/((24k+2)^2)-(2560)/((24k+3)^2)+(2560)/((24k+4)^2)-(1280)/((24k+5)^2)-(2048)/((24k+6)^2)-(512)/((24k+7)^2)-(832)/((24k+9)^2)-(512)/((24k+10)^2)+(128)/((24k+11)^2)-(128)/((24k+12)^2)-(80)/((24k+13)^2)+(16)/((24k+14)^2)+(40)/((24k+15)^2)+(20)/((24k+17)^2)+(40)/((24k+18)^2)+8/((24k+19)^2)+(10)/((24k+20)^2)+(13)/((24k+21)^2)+1/((24k+22)^2)-2/((24k+23)^2)]
(32)

(E. W. Weisstein, Feb. 25, 2006).

A rapidly converging Zeilberger-type sum due to A. Lupas is given by

K=1/(64)sum_(k=1)^infty((-1)^(k-1)2^(8k)(40k^2-24k+3)[(2k)!]^3(k!)^2)/(k^3(2k-1)[(4k)!]^2)
(33)

(Lupas 2000), which is used to calculate K in the Wolfram Language. This can also be written as

K=-1/(64)sum_(k=1)^infty((-2^8)^k(40k^2-23k+3))/(k^3(2k-1)(4k; 2k)^2(2k; k))
(34)

(Campbell 2022). Using the WZ method, Guillera (2019) obtained the formula

K=1/2sum_(k=0)^infty((-2^6)^k(4k+3))/((2k+1)^3(2k; k)^3)
(35)

(Guillera 2019, Campbell 2022). In addition, Campbell (2022) showed

K=-1/2-1/(16)sum_(k=1)^infty((-2^8)^k(40k^2+4k-1))/(k^2(4k+1)(2k; k)(4k; 2k)^2).
(36)

Catalan's constant is also given by the integrals

K=int_0^1(tan^(-1)xdx)/x
(37)
=int_0^13/xtan^(-1)[(x(1-x))/(2-x)]dx
(38)
=-int_0^1(lnxdx)/(1+x^2)
(39)
=1/2int_0^1K(k)dk
(40)
=-int_0^(pi/2)ln[2sin(1/2t)]dt
(41)
=int_0^(pi/4)ln(cotx)dx
(42)
=1/2int_0^(pi/2)xcscxdx
(43)
=pi/8int_(-infty)^infty(sechttanht)/tdt
(44)
=int_0^(pi/2)sinh^(-1)(sinx)dx
(45)
=1/2piln(1+sqrt(2))+int_0^(sinh^(-1)1)sin^(-1)(sinht)dt,
(46)

where (40) is from Mc Laughlin (2007; which corresponds to the 1/(-64)^k BBP-type formula), (41) is from Borwein et al. (2004, p. 106), (43) is from Glaisher (1877), (44) is from J. Borwein (pers. comm., Jul. 16, 2007), (45) is from Adamchik, and (46) is from W. Gosper (pers. comm., Jun. 11, 2008). Here, K(k) (not to be confused with Catalan's constant itself) is a complete elliptic integral of the first kind. Zudilin (2003) gives the unit square integral

K=1/8int_0^1int_0^1(dxdy)/((1-xy)sqrt(x(1-y))),
(47)

which is the analog of a double integral for zeta(2) due to Beukers (1979).

In terms of the trigamma function psi_1(x),

K=1/(16)psi_1(1/4)-1/(16)psi_1(3/4)
(48)
=1/8pi^2-1/8psi_1(3/4)
(49)
=1/(32)psi_1(1/8)+1/(32)psi_1(5/8)-1/8pi^2
(50)
=1/8pi^2-1/(32)psi_1(3/8)-1/(32)psi_1(7/8)
(51)
=1/(64)[psi_1(1/8)-psi_1(3/8)+psi_1(5/8)-psi_1(7/8)]
(52)
=1/(80)psi_1(5/(12))+1/(80)psi_1(1/(12))-1/(10)pi^2
(53)
=1/(10)pi^2-1/(80)psi_1(7/(12))-1/(80)psi_1((11)/(12))
(54)
=1/(160)[psi_1(1/(12))+psi_1(5/(12))-psi_1(7/(12))-psi_1((11)/(12))].
(55)

Catalan's constant also arises in products, such as

e^(-1/2+2K/pi)=lim_(n->infty)1/((4n+1)^(2n))product_(k=1)^n((4k-1)^(4k-1))/((4k-3)^(4k-3))
(56)

(Glaisher 1877).

Zudilin (2003) gives the continued fraction

K=((13)/2)/(q(0)+)(1^4·2^4·p(0)p(2))/(q(1)+)... ...((2n-1)^4(2n)^4p(n-1)p(n+1))/(q(n)+)...,
(57)

where

p(n)=20n^2-8n+1
(58)
q(n)=3520n^6+5632n^5+2064n^4-384n^3-156n^2+16n+7,
(59)

which is an analog of the continued fraction of Apéry's constant found by Apéry (1979).

See also

Catalan's Constant Approximations, Catalan's Constant Continued Fraction, Catalan's Constant Digits, Dirichlet Beta Function

Related Wolfram sites

http://functions.wolfram.com/Constants/Catalan/

Portions of this entry contributed by Jonathan Sondow (author's link)

Portions of this entry contributed by Oleg Marichev

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.Adamchik, V. "Integral and Series Representations for Catalan's Constant." http://www-2.cs.cmu.edu/~adamchik/articles/catalan.htm. Adamchik, V. "Thirty-Three Representations of Catalan's Constant." http://library.wolfram.com/infocenter/Demos/109/.Apéry, R. "Irrationalité de zeta(2) et zeta(3)." Astérisque 61, 11-13, 1979.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 551-552, 1985.Beukers, F. "A Note on the Irrationality of zeta(2) and zeta(3)." Bull. London Math. Soc. 11, 268-272, 1979.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Campbell, J. M. "WZ Proofs of Identities From Chu and Kiliç, With Applications." Appl. Math. E-Notes, 22, 354-361, 2022.Catalan, E. "Sur la transformation des series, et sur quelques integrales definies." Mémoires in 4 de l'Academie royale de Belgique, 1865.Catalan, E. "Recherches sur la constant G, et sur les integrales euleriennes." Mémoires de l'Academie imperiale des sciences de Saint-Pétersbourg, Ser. 7, 31, 1883.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. R. "Catalan's Constant." §1.7 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 53-59, 2003.Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.Glaisher, J. W. L. "On a Numerical Continued Product." Messenger Math. 6, 71-76, 1877.Gosper, R. W. "A Calculus of Series Rearrangements." In Algorithms and Complexity: New Directions and Recent Results. Proc. 1976 Carnegie-Mellon Conference (Ed. J. F. Traub). New York: Academic Press, pp. 121-151, 1976.Gosper, R. W. "Thought for Today." math-fun@cs.arizona.edu posting, Aug. 8, 1996.Guillera, J. "a New Formula for Computing the Catalan Constant." May 8, 2019. http://anamat.unizar. es/jguillera/other/catalan-form.pdf.Lupas, A. "Formulae for Some Classical Constants." In Proceedings of ROGER-2000. 2000. http://www.lacim.uqam.ca/~plouffe/articles/alupas1.pdf.Mc Laughlin, J. "An Integral for Catalan's Constant." 27 Sep 2007. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0709&L=nmbrthry&T=0&P=3444.Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany: Halle, pp. 105 and 151, 1909.Plouffe, S. "Table of Current Records for the Computation of Constants." http://pi.lacim.uqam.ca/eng/records_en.html.Rivoal, T. and Zudilin, W. "Diophantine Properties of Numbers Related to Catalan's Constant." Math. Ann. 326, 705-721, 2003. http://www.mi.uni-koeln.de/~wzudilin/beta.pdf.Sloane, N. J. A. Sequence A006752/M4593 in "The On-Line Encyclopedia of Integer Sequences."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.Zudilin, W. "An Apéry-Like Difference Equation for Catalan's Constant." Electronic J. Combinatorics 10, No. 1, R14, 1-10, 2003. http://www.combinatorics.org/Volume_10/Abstracts/v10i1r14.html.

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Catalan's Constant

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Marichev, Oleg; Sondow, Jonathan; and Weisstein, Eric W. "Catalan's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CatalansConstant.html

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