Isotropic Tensor

Author

Eric W. Weisstein
September 20, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/IsotropicTensor.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Recurrence

V5

RSolve[{a[n]==(((n-1))/((n+1)))(2a[n-1]+3a[n-2]),a[0]==1,a[1]==0},a[n],n]

a[0]:=1
a[1]:=0
a[n_]:=a[n]=(((n-1))/((n+1)))(2a[n-1]+3a[n-2])

Array[a,10,0]

V6

RSolve[{a[n]==(((n-1))/((n+1)))(2a[n-1]+3a[n-2]),a[0]==1,a[1]==0},a[n],n]

V7

RSolve[{a[n]==(((n-1))/((n+1)))(2a[n-1]+3a[n-2]),a[0]==1,a[1]==0},a[n],n]

FunctionExpand[%]

Table[DifferenceRoot[Function[{𝓎,𝓃},{(-3-3 𝓃) 𝓎[𝓃]+(-2-2 𝓃) 𝓎[1+𝓃]+(3+𝓃) 𝓎[2+𝓃]==0,𝓎[0]==1,𝓎[1]==0}]][n],{n,0,20}]

FindGeneratingFunction[{1,0,1,1,3,6,15,36,91,232,603},x]

Series[1-((-1+2 x+((Sqrt[1-3 x])/(Sqrt[1+x])))/(2 (x^2)))x,{x,0,20}]

Together[1-((-1+2 x+((Sqrt[1-3 x])/(Sqrt[1+x])))/(2 (x^2)))x]//FullSimplify

FindGeneratingFunction[{0,1,1,3,6,15,36,91,232,603},x]

Series[((1-((Sqrt[1-3 x])/(Sqrt[1+x])))/(2 x)),{x,0,20}]

SeriesCoefficient[((1-((Sqrt[1-3 x])/(Sqrt[1+x])))/(2 x)),{x,0,n}]

Table[(1/2) (Binomial[0,1+n]-(((-3))^(1+n)) Binomial[(1/2),1+n] Hypergeometric2F1[-1-n,(1/2),(1/2)-n,-(1/3)]),{n,0,10}]

DifferenceRoot[Function[{𝓎,𝓃},{(-3-3 𝓃) 𝓎[𝓃]+(-2-2 𝓃) 𝓎[1+𝓃]+(3+𝓃) 𝓎[2+𝓃]==0,𝓎[0]==1,𝓎[1]==0}]][100]

(1/2) (-(((-3))^(1+n)) Binomial[(1/2),1+n] Hypergeometric2F1[-1-n,(1/2),(1/2)-n,-(1/3)])/.n->100

Sum[(1/2) (-(((-3))^(1+n)) Binomial[(1/2),1+n] Hypergeometric2F1[-1-n,(1/2),(1/2)-n,-(1/3)])(x^n),{n,0,∞}]

Table[Binomial[0,1+n],{n,0,10}]

FullSimplify[FunctionExpand[-(1/2) (((-3))^(1+n)) Binomial[(1/2),1+n] Hypergeometric2F1[-1-n,(1/2),(1/2)-n,-(1/3)]],n∈Integers&&n>0]

Table[(((((-1))^n) (3^(1+n)) (Sqrt[π]) Hypergeometric2F1Regularized[(1/2),-1-n,(1/2)-n,-(1/3)])/(4 Gamma[2+n])),{n,0,10}]

Sum[(((((-1))^n) (3^(1+n)) (Sqrt[π]) Hypergeometric2F1Regularized[(1/2),-1-n,(1/2)-n,-(1/3)])/(4 Gamma[2+n]))(x^n),{n,0,∞}]

Sum[(((((-1))^n) (3^(1+n)) (Sqrt[π]) Hypergeometric2F1Regularized[(1/2),-1-n,(1/2)-n,-(1/3)])/(4 Gamma[2+n]))(x^n),{n,0,10}]

Series[((1-((Sqrt[1-3 x])/(Sqrt[1+x])))/(2 x)),{x,0,10}]

GeneratingFunction[(((((-1))^n) (3^(1+n)) (Sqrt[π]) Hypergeometric2F1Regularized[(1/2),-1-n,(1/2)-n,-(1/3)])/(4 Gamma[2+n])),n,x]