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_(n=1)^infty((-1)^(n-1)2^(8n)(40n^2-24n+3)[(2n)!]^3(n!)^2)/(n^3(2n-1)[(4n)!]^2)
(33)

(Lupas 2000), and is used to calculate K in the Wolfram Language.

Catalan's constant is also given by the integrals

K=int_0^1(tan^(-1)xdx)/x
(34)
=int_0^13/xtan^(-1)[(x(1-x))/(2-x)]dx
(35)
=-int_0^1(lnxdx)/(1+x^2)
(36)
=1/2int_0^1K(k)dk
(37)
=-int_0^(pi/2)ln[2sin(1/2t)]dt
(38)
=int_0^(pi/4)ln(cotx)dx
(39)
=1/2int_0^(pi/2)xcscxdx
(40)
=pi/8int_(-infty)^infty(sechttanht)/tdt
(41)
=int_0^(pi/2)sinh^(-1)(sinx)dx
(42)
=1/2piln(1+sqrt(2))+int_0^(sinh^(-1)1)sin^(-1)(sinht)dt,
(43)

where (37) is from Mc Laughlin (2007; which corresponds to the 1/(-64)^k BBP-type formula), (38) is from Borwein et al. (2004, p. 106), (40) is from Glaisher (1877), (41) is from J. Borwein (pers. comm., Jul. 16, 2007), (42) is from Adamchik, and (43) 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))),
(44)

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)
(45)
=1/8pi^2-1/8psi_1(3/4)
(46)
=1/(32)psi_1(1/8)+1/(32)psi_1(5/8)-1/8pi^2
(47)
=1/8pi^2-1/(32)psi_1(3/8)-1/(32)psi_1(7/8)
(48)
=1/(64)[psi_1(1/8)-psi_1(3/8)+psi_1(5/8)-psi_1(7/8)]
(49)
=1/(80)psi_1(5/(12))+1/(80)psi_1(1/(12))-1/(10)pi^2
(50)
=1/(10)pi^2-1/(80)psi_1(7/(12))-1/(80)psi_1((11)/(12))
(51)
=1/(160)[psi_1(1/(12))+psi_1(5/(12))-psi_1(7/(12))-psi_1((11)/(12))].
(52)

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))
(53)

(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)+)...,
(54)

where

p(n)=20n^2-8n+1
(55)
q(n)=3520n^6+5632n^5+2064n^4-384n^3-156n^2+16n+7,
(56)

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

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