Catalan's Constant

Author

Eric W. Weisstein
September 26, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/CatalansConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CatalansConstant.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Notation

Catalan//TraditionalForm

Properties

Element[Catalan,#]&/@{Algebraics,Rationals}

Zudilin's Continued Fraction

p[n_]:=20(n^2)-8n+1
q[n_]:=3520(n^6)+5632(n^5)+2064(n^4)-384(n^3)-156(n^2)+16n+7

V6

FromGeneralContinuedFraction[{a0_,l_List?MatrixQ}]:=a0+Fold[#2[[1]]/(#1+#2[[2]])&,0,Reverse[l]]

((13/2)/(q[0]+(((1^4)(2^4)p[0]p[2])/(q[1]))))//N

V7

p[n_]:=20(n^2)-8n+1
q[n_]:=3520(n^6)+5632(n^5)+2064(n^4)-384(n^3)-156(n^2)+16n+7

13/2/(q[0]+ContinuedFractionK[(((2n-1))^4)(((2n))^4)p[n-1]p[n+1],q[n],{n,∞}])

N[13/2/(q[0]+ContinuedFractionK[(((2n-1))^4)(((2n))^4)p[n-1]p[n+1],q[n],{n,20}]),20]

N[Catalan,20]

Analytic Expressions

β(2)

DirichletBeta[2]

i(Subscript[χ, 2])(i)

-I LegendreChi[2,I]

Glaisher

π/24-π/2Log[Glaisher]+4π Derivative[1,0][Zeta][-1,1/4]

FullSimplify[%]

Sums

(∑_(k=0)^∞)((((-1))^k)/(((2 k+1))^2))

Sum[((((-1))^k)/(((2k+1))^2)),{k,0,∞}]

2 (∑_(k=0)^∞)(1/(((4 k+1))^2))-((π^2)/8)

Simplify[-Pi^2/8+2Sum[(1/(((4k+1))^2)),{k,0,∞}]]

((π^2)/8)-2 (∑_(k=0)^∞)(1/(((4 k+3))^2))

Simplify[Pi^2/8-2Sum[(1/(((4k+3))^2)),{k,0,∞}]]

(∑_(k=0)^∞)(1/(((4 k+1))^2))-(∑_(k=0)^∞)(1/(((4 k+3))^2))

Simplify[Sum[(1/(((4k+1))^2)),{k,0,∞}]-Sum[(1/(((4k+3))^2)),{k,0,∞}]]

(1/16) (∑_(m=1)^∞)(((m+1) ((3^m)-1) ζ(m+2))/(4^m))

(1/16)Sum[(m+1)(((3^m)-1)/(4^m))Zeta[m+2],{m,∞}]

1-(∑_(n=1)^∞)((n ζ(2 n+1))/(16^n))

Glaisher (1913)

1-Sum[((n Zeta[2n+1])/(16^n)),{n,∞}]

(1/64) (∑_n^∞)(((((-1))^(n-1)) ((2^(8 n)) (40 (n^2)-24 n+3) ((((2 n)!))^3) (((n!))^2)))/((n^3) (2 n-1) ((((4 n)!))^2)))

Lupas Zeilberger Sum

(1/64)Sum[(((-1)^(n-1) (2^(8 n) (40 n^2-24 n+3) ((2 n)!)^3 (n!)^2))/(n^3 (2 n-1) ((4 n)!)^2)),{n,∞}]

$Version

-(1/64) (∑_k^∞)(((((-(2^8)))^k) (40 (k^2)-24 k+3))/((k^3) (2 k-1) ((Binomial[4 k, 2 k])^2) (Binomial[2 k, k])))

bug(428327): Implement -(1/64) Sum[((-2^8)^k (40 k^2 - 24 k + 3))/(k^3 Binomial[4 k, 2 k]^2 Binomial[2 k, k] (2 k - 1)), {k, Infinity}] == Catalan

https://www.math.nthu.edu.tw/~amen/2022/AMEN-210618.pdf

-(1/64)Sum[(((((-(2^8)))^k)(40(k^2)-24k+3))/((k^3)((Binomial[4k,2k])^2)Binomial[2k,k](2k-1))),{k,∞}]

N[%]

Catalan//N

$Version

-(1/16) (∑_k^∞)(((((-(2^8)))^k) (40 (k^2)+4 k-1))/((k^2) (4 k+1) (Binomial[2 k, k]) ((Binomial[4 k, 2 k])^2)))-(1/2)

bug(428328): Implement  -(1/2) - 1/16 Sum[((-2^8)^k (40 k^2 + 4 k - 1))/(k^2 (4 k + 1) Binomial[2 k, k] Binomial[4 k, 2 k]^2), {k, Infinity}] == Catalan

https://www.math.nthu.edu.tw/~amen/2022/AMEN-210618.pdf

$Version

-(1/2)-(1/16)Sum[(((((-(2^8)))^k)(40(k^2)+4k-1))/((k^2)(4k+1)Binomial[2k,k]((Binomial[4k,2k])^2))),{k,∞}]

N[%]

Catalan//N

(∑_i^∞)(∑_j^∞)(((4^(i+j)) ((i-1)! (j-1)!))/((i+j)! ((2 i+2 j+1) (Binomial[2 (i+j), i+j]))))==8 (1-K)

Oloa

8.(1-Catalan)

Sum[(((i - 1.)!(j - 1)!)/((i + j)!))((4^(i+j))/((2i+2j+1)Binomial[2(i+j),i+j])), {i, 500}, {j, 500}]//N

V6

NSum[(((i - 1)!(j - 1)!)/((i + j)!))((4^(i+j))/((2i+2j+1)Binomial[2(i+j),i+j])), {i, Infinity}, {j, Infinity}]

Sum[(((i - 1)!(j - 1)!)/((i + j)!))((4^(i+j))/((2i+2j+1)Binomial[2(i+j),i+j])),{i,∞}]

V7

$Version

TimeConstrained[Sum[(((i - 1)!(j - 1)!)/((i + j)!))((4^(i+j))/((2i+2j+1)Binomial[2(i+j),i+j])), {i, Infinity}, {j, Infinity}],60]//Timing

(1/2) (∑_(n=0)^∞)(((((-(2^6)))^n) (4 n+3))/((((2 n+1))^3) ((Binomial[2 n, n])^3)))

bug(428326): Implement 1/2 Sum[((-2^6)^n (4 n + 3))/((2 n + 1)^3 Binomial[2 n, n]^3), {n, 0, Infinity}] == Catalan

https://www.math.nthu.edu.tw/~amen/2022/AMEN-210618.pdf

$Version

(1/2)Sum[(((((-(2^6)))^n)(4n+3))/((((2n+1))^3)((Binomial[2n,n])^3))),{n,0,∞}]

N[%]

Catalan//N

Products

(lim_(n->∞)) (Underoverscript[(1/(((4 n+1))^(2 n)))∏, k, n])((((4 k-1))^(4 k-1))/(((4 k-3))^(4 k-3)))==(e^(2 C/π-1/2))

Glaisher 1877

Exact

N[Exp[-1/2+2Catalan/Pi],20]

Finite

p=Table[(1/(((4n+1))^(2n)))Product[((((4k-1))^(4k-1))/(((4k-3))^(4k-3))),{k,n}],{n,5}]

Through[{Numerator,Denominator}[p]]

N[%,10]

v5.2

p1=(1/(((4n+1))^(2n)))Product[((((4k-1))^(4k-1))/(((4k-3))^(4k-3))),{k,n}]

Table[p1,{n,3}]//N

...wrong...

p2=(1/(((4n+1))^(2n)))((Product[(((4k-1))^(4k-1)),{k,n}])/(Product[(((4k-3))^(4k-3)),{k,n}]))

Table[p2,{n,3}]//N

v6

Product[((((4k-1))^(4k-1))/(((4k-3))^(4k-3))),{k,n}]

((Product[(((4k-1))^(4k-1)),{k,n}])/(Product[(((4k-3))^(4k-3)),{k,n}]))

V7

(1/(((4n+1))^(2n)))Product[((((4k-1))^(4k-1))/(((4k-3))^(4k-3))),{k,n}]

Limit[%,n->∞]//FullSimplify

(1/(((4n+1))^(2n)))((Product[(((4k-1))^(4k-1)),{k,n}])/(Product[(((4k-3))^(4k-3)),{k,n}]))

Limit[%,n->∞]//FullSimplify

Definite Integrals

(∫_0^1)(((tan^(-1))x)/x)dx

Integrate[ArcTan[x]/x,{x,0,1}]

(∫_0^1)((3 (tan^(-1))(((x (1-x))/(2-x))))/x)dx

NIntegrate[3 ArcTan[x(1-x)/(2-x)]/x,{x,0,1}]

Integrate[3 ArcTan[x(1-x)/(2-x)]/x,{x,0,1}]

TrigToExp[3 ArcTan[x(1-x)/(2-x)]/x]

Integrate[%,{x,0,1}]

FullSimplify[%]

N[{%,Catalan}]

-(∫_0^1)((log(x))/((x^2)+1))dx

-Integrate[((Log[x])/(1+(x^2))),{x,0,1}]

(1/2) (∫_0^1)K((k^2))dk

Integrate[EllipticK[(k^2)],{k,0,1}]/2

(∫_0^(π/2))log(2 sin((t/2)))dt

Integrate[Log[2Sin[t/2]],{t,0,π/2}]

(∫_0^(π/4))log(cot(x))dx

Integrate[Log[Cot[x]],{x,0,π/4}]

(1/2) (∫_0^(π/2))x csc(x)dx

Integrate[x Csc[x],{x,0,π/2}]/2

(π/8) (∫_(-∞)^∞)((sech(t) tanh(t))/t)dt

{NIntegrate[((Sech[t]Tanh[t])/t),{t,-∞,∞}],((8Catalan)/π)//N}

v5.2

Integrate[((Sech[t]Tanh[t])/t),{t,-∞,∞}]//Timing

v6.0

Integrate[((Sech[t]Tanh[t])/t),{t,-∞,∞}]//Timing

v6.1

Integrate[((Sech[t]Tanh[t])/t),{t,-∞,∞}]//Timing

(∫_0^(π/2))(sinh^(-1))(sin(x))dx

NIntegrate[ArcSinh[Sin[x]],{x,0,Pi/2}]

V5

Integrate[ArcSinh[Sin[x]],{x,0,Pi/2}]//Timing

V6.0

Integrate[ArcSinh[Sin[x]],{x,0,Pi/2}]//Timing

V7

Integrate[ArcSinh[Sin[x]],{x,0,Pi/2}]//Timing

(1/2) π ln(1+(Sqrt[2]))+(∫_0^((sinh^(-1))(1)))(sin^(-1))(sinh(t))dt

NIntegrate[ArcSin[Sinh[t]],{t,0,ArcSinh[1]}]

-Catalan+Pi*Log[1+Sqrt[2]]/2//N

Solve[i==-k+Pi*Log[1+Sqrt[2]]/2,k]

-NIntegrate[ArcSin[Sinh[t]],{t,0,ArcSinh[1]}]+(1/2)π Log[1+(Sqrt[2])]

FullSimplify[-Integrate[ArcSin[Sinh[t]],{t,0,ArcSinh[1]}]+(1/2)π Log[1+(Sqrt[2])]]//Timing

Double Integrals

(1/8) (∫_0^1)(∫_0^1)(1/((1-x y) (Sqrt[x (1-y)])))dydx

Zudilin (2003)

(1/8)Integrate[(1/((1-x y)(Sqrt[x(1-y)]))),{x,0,1},{y,0,1}]

Polygamma Identities

(1/16) (ψ^((1)))((1/4))-(1/16) (ψ^((1)))((3/4))

FullSimplify[1/16PolyGamma[1,1/4]-1/16PolyGamma[1,3/4]]

(1/32) (ψ^((1)))((1/8))+(1/32) (ψ^((1)))((5/8))-((π^2)/8)

V6.0

FullSimplify[1/32 PolyGamma[1,1/8]+1/32 PolyGamma[1,5/8]-Pi^2/8]

V6.1

FullSimplify[1/32 PolyGamma[1,1/8]+1/32 PolyGamma[1,5/8]-Pi^2/8]

(1/80) (ψ^((1)))((1/12))-((π^2)/10)+(1/80) (ψ^((1)))((5/12))

V6.0

FullSimplify[1/80PolyGamma[1,5/12]+1/80PolyGamma[1,1/12]-Pi^2/10]

V6.1

FullSimplify[1/80PolyGamma[1,5/12]+1/80PolyGamma[1,1/12]-Pi^2/10]

Search for More

Off[N::"meprec"]

PolyGammaIdentities[n_]:=Module[{v,lr,c},
v={Sequence@@(PolyGamma[1,#/n]&/@Select[Range[n-1],GCD[#,n]==1&]),1,Pi^2,Catalan};
lr=LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]],
{10^15v}]],30],10^-30]]//N;
Simplify[Solve[# /. Catalan->c,c]/.c->Catalan]&/@Select[
#==Chop[N[#,20]]&/@(#.v&/@Round/@Most/@lr[[Flatten[Position[Chop/@Last/@lr,0]]]]),!FreeQ[#,Catalan]&]
]

Do[Print[{i,PolyGammaIdentities[i]//Timing}],{i,30}]

PolyGammaIdentities2[n_]:=Module[{v,lr,c},
v={Sequence@@(PolyGamma[1,#/n]&/@Select[Range[n-1],GCD[#,n]==1&]),1,Catalan};
lr=LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]],
{10^15v}]],30],10^-30]]//N;
Simplify[Solve[# /. Catalan->c,c]/.c->Catalan]&/@Select[
#==Chop[N[#,20]]&/@(#.v&/@Round/@Most/@lr[[Flatten[Position[Chop/@Last/@lr,0]]]]),!FreeQ[#,Catalan]&]
]

Do[Print[{i,PolyGammaIdentities2[i]//Timing}],{i,30}]

BBP-Type Identities

<<Utilities`Simplify`

Monitor[Do[If[(b=FindBasis[Catalan,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16,16},{n,64},{p,1}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,11,16},{n,64},{p,3,3}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,13,16},{n,64},{p,3,3}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,3,16},{n,64},{p,3,3}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Catalan,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,12,16},{n,64},{p,3,3}],{3^k,n,p}];Print["DONE"]

Summary

BasisSum[#,k]&/@FindBasis[Catalan,-64,12,BasisPower->2]//TraditionalForm

{C==(3/64) (∑_(k=0)^∞)(1/(((-64))^k)) ((32/(((12 k+1))^2))-(32/(((12 k+2))^2))-(32/(((12 k+3))^2))-(8/(((12 k+5))^2))-(16/(((12 k+6))^2))-(4/(((12 k+7))^2))-(4/(((12 k+9))^2))-(2/(((12 k+10))^2))+(1/(((12 k+11))^2)))}

BasisSum[#,k]&/@FindBasis[Catalan,4096,24,BasisPower->2]//TraditionalForm

BasisSum[#,k]&/@{{Catalan,{-96,96,96,0,24,48,12,0,12,6,-3,0,64},-64,12,2},{Catalan,{2560,-7680,1280,0,2560,1920,-320,0,160,320,80,0,-40,120,80,0,10,-30,5,0,-1024},-1024,20,2}}//TraditionalForm

alist:={1,1,1,0,-1,-1,-1,0}
k[n_]:=Pi Log[2]/8+ Sum[alist[[Mod[i-1,8]+1]]/2^Floor[(i+1)/2]/i^2,{i,n}]
N[k[20]]

FullSimplify[(1/((2^((i+1)/2))(i^2)))+(1/((2^((i+1)/2))(((i+1))^2)))+(1/((2^((i+3)/2))(i+2)^2))+-(1/((2^((i+5)/2))(i+4)^2))-(1/((2^((i+5)/2))(i+5)^2))-(1/((2^((i+7)/2))(i+6)^2))]

Sum[(2^((1/2) (-7-i))) ((8/(i^2))+(8/(((1+i))^2))+(4/(((2+i))^2))-(2/(((4+i))^2))-(2/(((5+i))^2))-(1/(((6+i))^2))),{i,1,∞,8}]+Pi Log[2]/8

N[%,20]

Khinchin and Lévy Constant

KhinchinList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Rest@FoldList[Times,1,Rest[ContinuedFraction[Mod[x,1],n+1]]]]
KhinchinLevyList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Denominator/@FromContinuedFraction/@Reverse[NestList[Drop[#,-1]&,Rest[ContinuedFraction[Mod[x,1],n+1]],n]]]

<<Graphics`Colors`
<<Utilities`Typesetting`

With[{k=5000,c=Catalan},
g1=ListPlot[KhinchinList[c,k],PlotJoined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,((Subscript[a, 1])(Subscript[⋯a, n]))^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin's constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[2.685452001]}];
g2=ListPlot[KhinchinLevyList[c,k],PlotJoined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,(Subscript[q, n])^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin-Lévy constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[Exp[Pi^2/12/Log[2]]]}
];
]//Timing

Show[GraphicsArray[{g1,g2}]]