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 LegendreP

 http://functions.wolfram.com/07.08.06.0040.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "k"]], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "2"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", 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"]], ")"]], "\[Nu]"], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", "\[Nu]"]]], 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["2", " ", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Mu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "-", "1"]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", 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["\[Mu]", "+", "\[Nu]"]], "\[LessEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", 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RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "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]"]], 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29