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variants of this functions
JacobiP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Identities > Functional identities > Relations between contiguous functions > Recurrence relations





http://functions.wolfram.com/07.15.17.0020.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "-", "b"]], ")"]], " ", "z"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "-", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", "+", "2"]], ",", RowBox[List["b", "-", "2"]], ",", "z"]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]]]], ")"]]]]]]]]








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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "-", "b"]], ")"]], " ", "z"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "-", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", "+", "2"]], ",", RowBox[List["b", "-", "2"]], ",", "z"]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]]]]]]]]]


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





2007-05-02