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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/07.15.17.0006.01









  


  










Input Form





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










Standard Form





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










Rule Form





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










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





2001-10-29





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