Continued Fraction Constants

Author

Eric W. Weisstein
April 16, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

(Subscript[C, 1])==(Underoverscript[\[ContinuedFractionK], (k=1), ∞])(1/k)==((BesselI[1, 2])/(BesselI[0, 2]))

ContinuedFractionK[1,k, {k, n}]

ContinuedFractionK[1,k, {k, 1∞}]

A052119

ContinuedFractionK[N[1,100],k, {k, 1*^4}]

N[%,100]

Table[ContinuedFractionK[1,k, {k, n}],{n,10}]

Limit

Limit[((BesselI[1+n,-2] BesselK[1,2]+BesselI[1,2] BesselK[1+n,2])/(-BesselI[1+n,-2] BesselK[0,2]+BesselI[0,2] BesselK[1+n,2])),n->∞]

Recurrence

Numerator

A001053

Numerator[%]

RSolve[{a[n+1]==n a[n]+a[n-1],a[1]==0,a[2]==1},a[n],n]

FullSimplify[FunctionExpand[%]]

Table[2 ((((-1))^n) BesselI[n,2] BesselK[1,2]+BesselI[1,2] BesselK[n,2]),{n,10}]//N

Denominator

A001040

Denominator[terms]

RSolve[{a[n+1]==n a[n]+a[n-1],a[0]==0,a[1]==1},a[n],n]

FullSimplify[FunctionExpand[%]]

Table[-2 (((-1))^n) BesselI[n,2] BesselK[0,2]+2 BesselI[0,2] BesselK[n,2],{n,10}]//N

Ratio

r1=FullSimplify[(((BesselI[n,-2] BesselK[1,2]-BesselI[1,-2] BesselK[n,2])/(BesselI[2,-2] BesselK[1,2]-BesselI[1,-2] BesselK[2,2]))/((BesselI[n,-2] BesselK[0,2]-BesselI[0,-2] BesselK[n,2])/(BesselI[1,-2] BesselK[0,2]-BesselI[0,-2] BesselK[1,2])))]

r2=FullSimplify[FunctionExpand[%]]

Rationality

Element[BesselI[1,2]/BesselI[0,2],#]&/@{Algebraics,Rationals}

Formating

<<NumberTheory`ContinuedFractions`

ContinuedFractionForm[Range[0,10]]

(Subscript[C, 2])==(Underoverscript[\[ContinuedFractionK], (k=1), ∞])(k/k)==(1/(e-1))

ContinuedFractionK[k,k,{k,n}]

FullSimplify[%]

Table[((((((n+1)!)/(Subfactorial[n+1]))-1))^(-1)),{n,10}]

Table[ContinuedFractionK[k,k,{k,n}],{n,10}]

Through[{Numerator,Denominator}[%]]

Limit[(1/(-1+((Gamma[2+n])/(Subfactorial[1+n])))),n->∞]

ContinuedFractionK[k,k,{k,∞}]

N[(1/(-1+E)),100]

(Subscript[C, 3])==1+(Underoverscript[\[ContinuedFractionK], (k=1), ∞])(k/1)==((((Sqrt[((π e)/2)]) erfc((1/(Sqrt[2])))))^(-1))

ContinuedFractionK[k,1,{k,n}]

FunctionExpand[%]

A225435/A225436

Table[ContinuedFractionK[k,1,{k,n}],{n,10}]

Table[1+ContinuedFractionK[k,1,{k,n}],{n,10}]

FindGeneratingFunction[Table[1+ContinuedFractionK[k,1,{k,n}],{n,30}],x]//Timing

ContinuedFractionK[k,1,{k,∞}]

...wrong divergence message: bug(201877)...

A111129

N[1/(Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]]),100]

1+ContinuedFractionK[k,N[1,200],{k,5*^4}]//N[#,100]&

1+ContinuedFractionK[k,N[1,100],{k,1*^5}]//N[#,100]&

Numerator

Numerator[Table[ContinuedFractionK[k,1,{k,n}],{n,10}]]

Numerator[Table[1+ContinuedFractionK[k,1,{k,n}],{n,10}]]

FindGeneratingFunction[Numerator[Table[ContinuedFractionK[k,1,{k,n}],{n,30}]],x]//Timing

Claimed E.g.f. from OEIS; appears to be wrong:

egf=Series[(1/2)*(-2+E^((1/2)*z*(2+z))*(1+z)(2+Sqrt[2*E*Pi]*Erf[1/Sqrt[2]])-E^((1/2)*(1+z)^2)*Sqrt[2*Pi]*(1+z)*Erf[(1+z)/Sqrt[2]]),{z,0,10}]

CoefficientList[egf,z]Range[0,10]!

CoefficientList[egf,z]{1,1,2,3,12,30,180,1260,2016,45360,226800}

SeriesCoefficient[(1/2)(-2+E^((1/2)*z(2+z))(1+z)(2+Sqrt[2E Pi]Erf[1/Sqrt[2]])-E^((1/2)*(1+z)^2)Sqrt[2Pi](1+z)Erf[(1+z)/Sqrt[2]]),{z,0,n}]//Timing

TimeConstrained[SeriesCoefficient[(1/2)(-2+E^((1/2)*z(2+z))(1+z)(2+Sqrt[2E Pi]Erf[1/Sqrt[2]])-E^((1/2)*(1+z)^2)Sqrt[2Pi](1+z)Erf[(1+z)/Sqrt[2]]),{z,0,n}],60]//Timing//FullSimplify

Denominator

Denominator[Table[ContinuedFractionK[k,1,{k,n}],{n,10}]]

FindGeneratingFunction[Denominator[Table[ContinuedFractionK[k,1,{k,n}],{n,30}]],x]//Timing

Claimed E.g.f. from OEIS; appears to be wrong:

SeriesCoefficient[(1/2)(2+E^((1/2)(1+z)^2)Sqrt[2Pi](1+z)(-Erf[1/Sqrt[2]]+Erf[(1+z)/Sqrt[2]])),{z,0,n}]//FullSimplify

egf=Series[(1/2)(2+E^((1/2)(1+z)^2)Sqrt[2Pi](1+z)(-Erf[1/Sqrt[2]]+Erf[(1+z)/Sqrt[2]])),{z,0,10}]

CoefficientList[egf,z]{1,1,2,3,12,30,180,1260,2016,45360,226800}

Range[0,10]!

{1,1,2,6,24,120,720,5040,40320,362880,3628800}/{1,1,2,3,12,30,180,1260,2016,45360,226800}

(Subscript[C, 4])==1+(Underoverscript[\[ContinuedFractionK], (k=1), ∞])((2 k)/(2 k+1))==(1/((Sqrt[e])-1))

1+ContinuedFractionK[2k,2k+1,{k,∞}]

A113011

1+ContinuedFractionK[2k,2k+N[1,100],{k,(10^3)}]

Table[1+ContinuedFractionK[2k,2k+1,{k,n}],{n,10}]

Through[{Numerator,Denominator}[%]]

Finite c.f.

1+ContinuedFractionK[2k,2k+1,{k,n}]

Together[1-(((Sqrt[E]) Gamma[2+n]-2 Gamma[2+n,-(1/2)])/((Sqrt[E]) Gamma[2+n]-Gamma[2+n,-(1/2)]))]

FullSimplify[%]

Limit[(1/(-1+(((Sqrt[E]) Gamma[2+n])/(Gamma[2+n,-(1/2)])))),n->∞]

FullSimplify[%]

bug(326566)

Subtract@@{(1/(-1+(Sqrt[E ]))),((Sqrt[π])/(-(Sqrt[π])+(Sqrt[E π])))}//FullSimplify

Numerator

A113012

n1=Table[1+ContinuedFractionK[2k,2k+1,{k,n}],{n,0,20}]//Numerator

n2=Table[1/((((Sqrt[E]) Gamma[n+2])/(Gamma[n+2,-(1/2)]))-1),{n,0,20}]//Numerator

n3=Table[Sum[(-1)^k*1/(2k)!!,{k,0,n}],{n,21}]//Numerator

Sum[(-1)^k*1/(2k)!!,{k,0,n}]//FullSimplify

n4=Table[((1-(((2^(1+n)) (1+n) (Gamma[1+n]-Gamma[1+n,-(1/2)]))/((2 (1+n))!!)))/(Sqrt[E])),{n,21}]//FullSimplify//Numerator

n1==n2==n3

Denominator

A113013

d1=Table[1+ContinuedFractionK[2k,2k+1,{k,n}],{n,0,20}]//Denominator

d2=Table[1/((((Sqrt[E]) Gamma[n+2])/(Gamma[n+2,-(1/2)]))-1),{n,0,20}]//Denominator

d1==d2