Gegenbauer function: Representations through more general functions (formula 07.14.26.0009)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,lambda,z]

Representations through more general functions

Through Meijer G

Classical cases involving algebraic functions

http://functions.wolfram.com/07.14.26.0009.01

Input Form

(1 + z)^(\[Lambda] - 1/2) GegenbauerC[\[Nu], \[Lambda], 2 z + 1] == ((Cos[(\[Lambda] + \[Nu]) Pi] Pochhammer[2 \[Lambda], \[Nu]])/ (Cos[\[Lambda] Pi] Gamma[1/2 - \[Lambda]] Gamma[\[Nu] + 1])) MeijerG[{{1/2 - \[Lambda] - \[Nu], 1/2 + \[Lambda] + \[Nu]}, {}}, {{0}, {1/2 - \[Lambda]}}, z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Lambda]", "-", FractionBox["1", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], ",", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> λ </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Lambda]"]], ")"]], "\[Nu]"], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> λ </mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> ν </ci> </apply> <ci> λ </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <times /> <ci> cos </ci> <apply> <times /> <apply> <plus /> <ci> λ </ci> <ci> ν </ci> </apply> <pi /> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <ci> cos </ci> <apply> <times /> <ci> λ </ci> <pi /> </apply> </apply> <apply> <times /> <ci> Γ </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> <apply> <times /> <ci> Γ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> λ </ci> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> </list> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </list> </list> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z_"]], ")"]], RowBox[List["\[Lambda]_", "-", FractionBox["1", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", RowBox[List[RowBox[List["2", " ", "z_"]], "+", "1"]]]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], ",", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Lambda]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]]]]]]

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

2001-10-29

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