Chebshyev Polynomial of the Second Kind

Author

Eric W. Weisstein
March 22, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/ChebyshevPolynomialoftheSecondKind.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

poly=ChebyshevU;

poly[#,x]&/@Range[0,10]//ColumnForm//TraditionalForm

DeleteCases[CoefficientList[poly[#,x],x],0]&/@Range[0,10]

Flatten[%]

Plot

Plot[Evaluate[Table[ChebyshevU[n,x],{n,5}]],{x,-1,1},PlotStyle->{Red,Yellow,Green,Blue,Violet}]

Table

Table[{n,ChebyshevU[n,x]},{n,0,5}]//TableForm//TraditionalForm

Flatten[Table[CoefficientList[ChebyshevU[n,x],x],{n,0,5}]]

Sums

(∑_(n=0)^∞)(t^n) (Subscript[U, n])(x)==(1/((t^2)-2 t x+1))

Sum[ChebyshevU[n,x](t^n),{n,0,∞}]

Series[%,{t,0,5}]

SeriesCoefficient[(1/(1+(t^2)-2 t x)),{t,0,n}]//InputForm

GeneratingFunction[ChebyshevU[n,x],n,t]

Derivation

Collect[Expand[Collect[Series[(1/(1+(t^2)-2 t x)),{t,0,5}],t]],t]

Simplify[D[(1/(1+(t^2)-2 t x)),t]]

Simplify[-((2 (t-x))/(((1+(t^2)-2 t x))^2))t+(1/(1+(t^2)-2 t x))]

Collect[Expand[Collect[Series[((1-(t^2))/(((1+(t^2)-2 t x))^2)),{t,0,5}],t]],t]

Table[n+1,{n,5}]

Table[ChebyshevT[n,x],{n,5}]

Table[ChebyshevU[n,x](n+1),{n,5}]

(∑_(r=0)^(⌊n/2⌋))(((-1))^r) (Binomial[n-r, r]) (((2 x))^(n-2 r))==(Subscript[U, n])(x)

Sum[(((-1))^r)Binomial[n-r,r](((2x))^(n-2r)),{r,0,Floor[n/2]}]

FullSimplify[%]

Table[(1/2) ((((x-(Sqrt[-1+(x^2)])))^n)+(((x+(Sqrt[-1+(x^2)])))^n)+((x (-(((x-(Sqrt[-1+(x^2)])))^n)+(((x+(Sqrt[-1+(x^2)])))^n)))/(Sqrt[-1+(x^2)]))),{n,5}]//Expand

(∑_(m=0)^(⌊n/2⌋))((((x^2)-1))^m) (Binomial[n+1, 2 m+1]) (x^(n-2 m))==(Subscript[U, n])(x)

Sum[Binomial[n+1,2m+1](x^(n-2m))((((x^2)-1))^m),{m,0,Floor[n/2]}]

Table[(1/(2 (Sqrt[-1+(x^2)])))(-x (((x-(Sqrt[-1+(x^2)])))^n)+(Sqrt[-1+(x^2)]) (((x-(Sqrt[-1+(x^2)])))^n)+x (((x+(Sqrt[-1+(x^2)])))^n)+(Sqrt[-1+(x^2)]) (((x+(Sqrt[-1+(x^2)])))^n)),{n,5}]//Expand

Table[ChebyshevU[n,x],{n,5}]

(∑_(n=0)^∞)n (t^(n-1)) (Subscript[U, n])(x)==((2 (x-t))/(((1-2 ((t^2)+t) x))^2))

Sum[n ChebyshevU[n,x](t^(n-1)),{n,0,∞}]

(∑_(n=0)^∞)n (t^n) (Subscript[U, n])(x)==((2 t x-2 (t^2))/(((1-2 ((t^2)+t) x))^2))

Sum[n ChebyshevU[n,x](t^n),{n,0,∞}]

(∑_(n=0)^∞)(n+1) (t^n) (Subscript[U, n])(x)==((((1-t))^2)/((((t^2)-2 t x+1))^2))

Sum[(n+1)ChebyshevU[n,x](t^n),{n,0,∞}]

Products

Table[FullSimplify[FunctionExpand[2^n Product[x-Cos[k π/(n+1)],{k,n}]]]-ChebyshevU[n,x],{n,0,5}]//Timing

(2^n) (∏_k^n)(x-cos(((π k)/(n+1))))==(Subscript[U, n])(x)

V6

2^n Product[x-Cos[k π/(n+1)],{k,n}]

V7

2^n Product[x-Cos[k π/(n+1)],{k,n}]

Identities

(Subscript[U, n])(x)==(((n+1) (Subscript[P, n]^((1/2,1/2)))(x))/((Subscript[P, n]^((1/2,1/2)))(1)))

FullSimplify[(n+1)((JacobiP[n,1/2,1/2,x])/(JacobiP[n,1/2,1/2,1]))]

(Subscript[U, n])(x)==(n+1) (Hypergeometric2F1[-n, n+2, (3/2), ((1-x)/2)])

FullSimplify[(n+1)Hypergeometric2F1[-n,n+2,(3/2),(1/2)(1-x)]]

FullSimplify[(n+1)Hypergeometric2F1[-n,n+2,(3/2),(1/2)(1-x)]-ChebyshevU[n,x]]

Subtract@@@Table[{(n+1)Hypergeometric2F1[-n,n+2,(3/2),(1/2)(1-x)],ChebyshevU[n,x]}/.{n->RandomReal[{0,10}],x->RandomComplexDisk[]},{100}]//Chop//Union

Subtract@@@Table[{(n+1)Hypergeometric2F1[-n,n+2,(3/2),(1/2)(1-x)],ChebyshevU[n,x]}/.{n->RandomComplexDisk[],x->RandomComplexDisk[]},{100}]//Chop//Union

Determinant Identity

ChebyshevUDet[n_]:=SparseArray[{{i_,i_}->2x,{i_,j_}/;Abs[i-j]==1->1},{n,n}]

chebDet[n_,x_]:=Module[{i,c=2x IdentityMatrix[n]},
Do[
c[[i,i+1]]=c[[i+1,i]]=1,
{i,n-1}];
c
]

ChebyshevUProd[n_,x_]:=Product[2(x-Cos[k Pi/(n+1)]),{k,n}]

ChebyshevUDet[5]//Normal//MatrixForm

Det/@Table[ChebyshevUDet[n,x],{n,5}]

Expand[Table[ChebyshevUProd[n,x],{n,5}]]

Table[ChebyshevU[n,x],{n,5}]