Gegenbauer Polynomial

Author

Eric W. Weisstein
September 6, 2008

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

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

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

Notation

TraditionalForm[GegenbauerC[μ,ν,x]]

Enumeration

Table[GegenbauerC[i,λ,x],{i,0,3}]

Collect[CoefficientList[Series[1/(1-2x t+t^2)^λ,{t,0,4}],t],x]//ColumnForm

Generating Function

Sum[GegenbauerC[n,λ,x](t^n),{n,0,∞}]

GeneratingFunction[GegenbauerC[n,λ,x],n,t]

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

FullSimplify[%]

In Terms of other functions

JacobiP

Ultraspherical[n_,λ_,x_]:=Gamma[λ+1/2]/Gamma[2λ]Gamma[n+2λ]/Gamma[n+λ+1/2]JacobiP[n,λ-1/2,λ-1/2,x]

FunctionExpand[GegenbauerC[n,λ,x]]

FullSimplify[Ultraspherical[n,λ,x]]

Hypergeometric2F1

UltrasphericalF[n_,λ_,x_]:=Binomial[n+2λ-1,n]Hypergeometric2F1[-n,n+2λ,λ+1/2,(1-x)/2]

FunctionExpand[GegenbauerC[n,λ,x]]

FunctionExpand[UltrasphericalF[n,λ,x]]

FullSimplify[UltrasphericalF[n,λ,x]]

FullSimplify[GegenbauerC[n,λ,x]-UltrasphericalF[n,λ,x]]

Subtract@@@Table[{GegenbauerC[n,λ,x],UltrasphericalF[n,λ,x]},{n,-5,5}]//FullSimplify//Union

Subtract@@@Table[{GegenbauerC[n,λ,x],UltrasphericalF[n,λ,x]}/.{n->RandomComplex[],λ->RandomComplex[],x->RandomComplex[]},{10}]//FullSimplify//Chop//Union

Integral

Integrate[(1-x^2)^(λ-1/2)GegenbauerC[n,λ,x]^2,{x,-1,1}]

Integrate[(1-x^2)^(λ-1/2)Ultraspherical[n,λ,x]^2,{x,-1,1}]

Special Values of λ

FunctionExpand[GegenbauerC[1/2,n,x]]

FullSimplify[FunctionExpand[UltrasphericalF[1/2,n,x]]]

FullSimplify[FunctionExpand[UltrasphericalF[3/2,n,x]]]

FullSimplify[Table[GegenbauerC[1/2,n,x]-c[n]LegendreP[n,x],{n,0,3}]]

FullSimplify[Table[GegenbauerC[1,n,x]-c[n]ChebyshevU[n,x],{n,0,3}]]

Differential Equation

DSolve[D[c[n,λ,x],x]==2λ c[n-1,λ+1,x],c[n,λ,x],{n,λ}]

Recurrence Equation

GegenbauerC[#,λ,x]&/@{0,1}

RSolve[{n c[n,λ,x]==2(n+λ-1)x c[n-1,λ,x]-(n+2λ-2)c[n-2,λ,x],c[0,λ,x]==1,c[1,λ,x]==2x λ},c[n,λ,x],n]