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variants of this functions
JacobiP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/07.15.17.0013.01









  


  








Input Form





JacobiP[\[Nu], a, b, -z] == Csc[b Pi] Sin[(b + \[Nu]) Pi] JacobiP[\[Nu], b, a, z] - (2^b Csc[b Pi] Csc[(a + \[Nu]) Pi] Sin[\[Nu] Pi] Sin[(a + b + \[Nu]) Pi] JacobiP[-1 - a - \[Nu], -b, a, z])/(1 - z)^b








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "b", ",", "a", ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "\[Nu]"]], ",", RowBox[List["-", "b"]], ",", "a", ",", "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> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mi> b </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <csc /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> &#957; </ci> </apply> <pi /> </apply> </apply> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> b </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> <apply> <csc /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> &#957; </ci> </apply> <pi /> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> </apply> <pi /> </apply> </apply> <apply> <ci> JacobiP </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> a </ci> <ci> z </ci> </apply> </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