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 LegendreP

 http://functions.wolfram.com/07.09.06.0037.01

 Input Form

 LegendreP[\[Nu], \[Mu], 3, z] == ((2^\[Nu] Gamma[1/2 + \[Nu]])/(Sqrt[Pi] Gamma[1 - \[Mu] + \[Nu]])) (z - 1)^(\[Nu] - \[Mu]/2) (z + 1)^(\[Mu]/2) Sum[((Pochhammer[\[Mu] - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! Pochhammer[-2 \[Nu], k])) (2/(1 - z))^k, {k, 0, \[Nu] - \[Mu]}] - (((-1)^(\[Nu] - \[Mu]) 2^(\[Nu] + 1) Gamma[\[Mu] + \[Nu] + 1])/ (Gamma[-\[Nu]] Gamma[2 \[Nu] + 2])) (z - 1)^(-\[Mu]/2 - \[Nu] - 1) (z + 1)^(\[Mu]/2) Hypergeometric2F1[\[Nu] + 1, \[Mu] + \[Nu] + 1, 2 \[Nu] + 2, 2/(1 - z)] /; Element[2 \[Nu] + 1, Integers] && 2 \[Nu] + 1 >= 0 && Element[\[Nu] - \[Mu], Integers] && -\[Nu] <= \[Mu] <= \[Nu] + 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Nu]", "-", RowBox[List["\[Mu]", "/", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", "\[Mu]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "\[Mu]"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]], "-", "\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]]]]]], " ", "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], "\[LessEqual]", "\[Mu]", "\[LessEqual]", RowBox[List["\[Nu]", "+", "1"]]]]]]]]]]

 MathML Form

 𝔓 ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) 2 ν Γ ( ν + 1 2 ) π Γ ( ν - μ + 1 ) ( z - 1 ) ν - μ / 2 ( z + 1 ) μ / 2 k = 0 ν - μ ( μ - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k - ( - 1 ) ν - μ 2 ν + 1 Γ ( μ + ν + 1 ) Γ ( - ν ) Γ ( 2 ν + 2 ) ( z - 1 ) - μ / 2 - ν - 1 ( z + 1 ) μ / 2 2 F 1 ( ν + 1 , μ + ν + 1 ; 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; 2 ν + 1 ν - μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] - ν μ ν + 1 Condition Subscript 𝔓 ν μ HoldComplete LegendreP 2 z 2 ν Gamma ν 1 2 1 2 Gamma ν -1 μ 1 -1 z -1 ν -1 μ 2 -1 z 1 μ 2 -1 k 0 ν -1 μ Pochhammer μ -1 ν k Pochhammer -1 ν k k Pochhammer -2 ν k -1 2 1 -1 z -1 k -1 -1 ν -1 μ 2 ν 1 Gamma μ ν 1 Gamma -1 ν Gamma 2 ν 2 -1 z -1 -1 μ 2 -1 -1 ν -1 z 1 μ 2 -1 Hypergeometric2F1 ν 1 μ ν 1 2 ν 2 2 1 -1 z -1 2 ν 1 ν -1 μ -1 ν μ ν 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", "\[Mu]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "\[Mu]"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], "-", "\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], "\[LessEqual]", "\[Mu]", "\[LessEqual]", RowBox[List["\[Nu]", "+", "1"]]]]]]]]]]]]

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

 2001-10-29