Gegenbauer polynomials: Specific values (formula 05.09.03.0018)

GegenbauerC

$Mathematica Notation$

Polynomials

GegenbauerC[n,lambda,z]

Specific values

Specialized values

For fixed lambda, z

http://functions.wolfram.com/05.09.03.0018.01

Input Form

GegenbauerC[8, \[Lambda], z] == (2/315) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (\[Lambda] + 3) (\[Lambda] + 4) (\[Lambda] + 5) (\[Lambda] + 6) (\[Lambda] + 7) z^8 - (4/45) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (\[Lambda] + 3) (\[Lambda] + 4) (\[Lambda] + 5) (\[Lambda] + 6) z^6 + (1/3) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (\[Lambda] + 3) (\[Lambda] + 4) (\[Lambda] + 5) z^4 - (1/3) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (\[Lambda] + 3) (\[Lambda] + 4) z^2 + (1/24) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (\[Lambda] + 3)

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mi> C </mi> <mn> 8 </mn> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 2 </mn> <mn> 315 </mn> </mfrac> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 4 </mn> <mn> 45 </mn> </mfrac> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <cn type='integer'> 8 </cn> </apply> <ci> λ </ci> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 2 <sep /> 315 </cn> <ci> λ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 4 <sep /> 45 </cn> <ci> λ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <ci> λ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <ci> λ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 24 </cn> <ci> λ </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

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