Associated Legendre polynomials: Identities (formula 05.07.17.0009)

LegendreP

$Mathematica Notation$

Polynomials

LegendreP[n,mu,2,z]

Identities

Functional identities

Expansion with respect to parameters

http://functions.wolfram.com/05.07.17.0009.01

Input Form

LegendreP[Subscript[n, 1], Subscript[\[Mu], 1], z] LegendreP[Subscript[n, 2], Subscript[\[Mu], 2], z] == (-1)^Subscript[\[Mu], 1] Sqrt[((Subscript[n, 1] + Subscript[\[Mu], 1])! (Subscript[n, 2] + Subscript[\[Mu], 2])!)/ ((Subscript[n, 1] - Subscript[\[Mu], 1])! (Subscript[n, 2] - Subscript[\[Mu], 2])!)] Sum[If[k >= Abs[Subscript[\[Mu], 1] - Subscript[\[Mu], 2]] && EvenQ[k + Subscript[n, 1] + Subscript[n, 2]], (-1)^(-Subscript[\[Mu], 1] + Subscript[\[Mu], 2]) (2 k + 1) ThreeJSymbol[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {k, 0}] ThreeJSymbol[{Subscript[n, 1], -Subscript[\[Mu], 1]}, {Subscript[n, 2], Subscript[\[Mu], 2]}, {k, Subscript[\[Mu], 1] - Subscript[\[Mu], 2]}] Sqrt[(k - Subscript[\[Mu], 2] + Subscript[\[Mu], 1])!/ (k + Subscript[\[Mu], 2] - Subscript[\[Mu], 1])!] LegendreP[k, -Subscript[\[Mu], 1] + Subscript[\[Mu], 2], z], 0], {k, Abs[Subscript[n, 1] - Subscript[n, 2]], Subscript[n, 1] + Subscript[n, 2]}] /; Element[Subscript[\[Mu], 1], Integers] && Element[Subscript[\[Mu], 2], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["\[Mu]", "1"], ",", "z"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["\[Mu]", "2"], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["\[Mu]", "1"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["\[Mu]", "1"]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "2"], "+", SubscriptBox["\[Mu]", "2"]]], ")"]], "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["\[Mu]", "1"]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "2"], "-", SubscriptBox["\[Mu]", "2"]]], ")"]], "!"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["n", "2"]]], "]"]]]], RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"]]]], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["k", "\[GreaterEqual]", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["\[Mu]", "1"], "-", SubscriptBox["\[Mu]", "2"]]], "]"]]]], "\[And]", RowBox[List["EvenQ", "[", RowBox[List["k", "+", SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"]]], "]"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", SubscriptBox["\[Mu]", "1"]]], "+", SubscriptBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "2"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "0"]], "}"]]]], "]"]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", RowBox[List["-", SubscriptBox["\[Mu]", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", RowBox[List[SubscriptBox["\[Mu]", "1"], "-", SubscriptBox["\[Mu]", "2"]]]]], "}"]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["\[Mu]", "2"], "+", SubscriptBox["\[Mu]", "1"]]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", SubscriptBox["\[Mu]", "2"], "-", SubscriptBox["\[Mu]", "1"]]], ")"]], "!"]]]], " ", RowBox[List["LegendreP", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", SubscriptBox["\[Mu]", "1"]]], "+", SubscriptBox["\[Mu]", "2"]]], ",", "z"]], "]"]]]], ",", "0"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Mu]", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["\[Mu]", "2"], "\[Element]", "Integers"]]]]]]]]

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> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </msup> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mrow> <mi> k </mi> <mo> ≥ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> EvenQ </mi> <mo> [ </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> ] </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["(", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext>   </mtext> <mtable> <mtr> <mtd> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mtext>   </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[")", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["(", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext>   </mtext> <mtable> <mtr> <mtd> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </mrow> </mtd> <mtd> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <mrow> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> </mtd> </mtr> </mtable> <mtext>   </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[")", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> k </mi> <mrow> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> μ </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> μ </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> LegendreP </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </uplimit> <apply> <ci> If </ci> <apply> <and /> <apply> <geq /> <ci> k </ci> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> EvenQ </ci> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> ThreeJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </list> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </list> <list> <ci> k </ci> <cn type='integer'> 0 </cn> </list> </apply> <apply> <ci> ThreeJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <ci> k </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </list> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> k </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> ℤ </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <ci> ℤ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18

LegendreP[n,z]
LegendreP[nu,z]
LegendreP[nu,mu,z]
LegendreP[nu,mu,2,z]
LegendreP[nu,mu,3,z]