Gegenbauer polynomials: Transformations (formula 05.09.16.0004)

GegenbauerC

$Mathematica Notation$

Polynomials

GegenbauerC[n,lambda,z]

Transformations

Addition formulas

http://functions.wolfram.com/05.09.16.0004.01

Input Form

GegenbauerC[n, \[Alpha], Cos[Subscript[\[Theta], 0]]] == Sum[(((Pochhammer[\[Alpha], k] (n - k)!) Sin[\[Theta]]^k Sin[\[CurlyTheta]]^k)/(Pochhammer[\[Alpha] - 1/2, k] Pochhammer[2 \[Alpha] + 2 k, n - k])) GegenbauerC[n - k, \[Alpha] + k, Cos[\[Theta]]] GegenbauerC[n - k, \[Alpha] + k, Cos[\[CurlyTheta]]] GegenbauerC[k, \[Alpha] - 1/2, Cos[\[Phi]]], {k, 0, n}] /; Element[n, Integers] && n >= 0 && Cos[Subscript[\[Theta], 0]] == Cos[\[Theta]] Cos[\[CurlyTheta]] + Sin[\[Theta]] Sin[\[CurlyTheta]] Cos[\[Phi]]

Standard Form

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MathML Form

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</mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> θ </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> θ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϕ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> θ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> α </ci> </apply> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <ci> α </ci> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> α </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <sin /> <ci> θ </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <sin /> <ci> ϑ </ci> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> α </ci> <ci> k </ci> </apply> </apply> <apply> <cos /> <ci> θ </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> α </ci> <ci> k </ci> </apply> </apply> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <ci> ϕ </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <eq /> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> θ </ci> </apply> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> <apply> <times /> <apply> <cos /> <ci> ϕ </ci> </apply> <apply> <sin /> <ci> θ </ci> </apply> <apply> <sin /> <ci> ϑ </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18

GegenbauerC[nu,z]
GegenbauerC[nu,lambda,z]