Jacobi function: Identities (formula 07.15.17.0007)

JacobiP

$Mathematica Notation$

Hypergeometric Functions

JacobiP[nu,a,b,z]

Identities

Functional identities

Relations between contiguous functions

Normalized recurrence relation

http://functions.wolfram.com/07.15.17.0007.01

Input Form

z p[\[Nu], z] == p[\[Nu] + 1, z] + ((b^2 - a^2)/((a + b + 2 \[Nu]) (a + b + 2 \[Nu] + 2))) p[\[Nu], z] + ((4 \[Nu] (a + \[Nu]) (b + \[Nu]) (a + b + \[Nu]))/ ((a + b + 2 \[Nu] - 1) (a + b + 2 \[Nu])^2 (a + b + 2 \[Nu] + 1))) p[\[Nu] - 1, z] /; p[\[Nu], z] == ((2^\[Nu] Gamma[\[Nu] + 1])/Pochhammer[a + b + \[Nu] + 1, \[Nu]]) JacobiP[\[Nu], a, b, z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "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["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], " "]]], RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["4", "\[Nu]", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " "]]], RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "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> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </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> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> P </mi> <mi> ν </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'> <mrow> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </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> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> P </mi> <mi> ν </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> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["z_", " ", RowBox[List["p", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["a", "2"]]], ")"]], " ", RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]]]]]]]]]]

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

2001-10-29

JacobiP[n,a,b,z]