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 LegendreP

 http://functions.wolfram.com/07.08.06.0003.01

 Input Form

 LegendreP[\[Nu], \[Mu], 2, z] == ((\[Mu] z)/2) Sum[(k + 1)! Pochhammer[1 + \[Mu]/2, k] z^k HypergeometricPFQ[{-(\[Mu]/2)}, {}, -z] HypergeometricPFQRegularized[ {-\[Nu], 1 + \[Nu], 1 - \[Mu]/2}, {k + 2, 1 - \[Mu]}, -(z/2)], {k, 0, Infinity}] + Gamma[1 - \[Mu]/2] Sum[Pochhammer[-(\[Mu]/2), k] (-z)^k HypergeometricPFQRegularized[ {{-\[Nu], 1 + \[Nu], 1 - \[Mu]/2}, {1}, {}}, {{1, 1 - \[Mu]}, {1 - \[Mu]/2}, {}}, 1/2, -(z/2)], {k, 0, Infinity}]

 Standard Form

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

 MathML Form

 P TagBox["P", LegendreP] ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) μ z 2 k = 0 ( k + 1 ) ! ( μ 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Mu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z k 1 F 0 ( - μ 2 ; ; - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["0", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] 3 F ~ 2 ( - ν , ν + 1 , 1 - μ 2 ; k + 2 , 1 - μ ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + Γ ( 1 - μ 2 ) k = 0 ( - μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "k"], Pochhammer] ( - z ) k F ~ 2 1 0 3 1 0 ( - ν , ν + 1 , 1 - μ 2 ; 1 ; ; 1 , 1 - μ ; 1 - μ 2 ; ; 1 2 , - z 2 ) FormBox RowBox RowBox SubsuperscriptBox TagBox P LegendreP ν μ ( TagBox z HoldComplete LegendreP 2 ) RowBox RowBox FractionBox RowBox μ z 2 RowBox UnderoverscriptBox RowBox k = 0 RowBox RowBox RowBox ( RowBox k + 1 ) ! TagBox SubscriptBox RowBox ( RowBox FractionBox μ 2 + 1 ) k Pochhammer SuperscriptBox z k TagBox TagBox RowBox RowBox SubscriptBox FormBox 1 TraditionalForm SubscriptBox F FormBox 0 TraditionalForm RowBox ( RowBox TagBox TagBox TagBox RowBox - FractionBox μ 2 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox RowBox - z HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ TagBox TagBox RowBox RowBox SubscriptBox FormBox 3 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox 2 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox - ν HypergeometricPFQRegularized Rule Editable , TagBox RowBox ν + 1 HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - FractionBox μ 2 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox k + 2 HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - μ HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox - FractionBox z 2 HypergeometricPFQRegularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQRegularized + RowBox RowBox Γ ( RowBox 1 - FractionBox μ 2 ) RowBox UnderoverscriptBox RowBox k = 0 ErrorBox RowBox TagBox SubscriptBox RowBox ( RowBox - FractionBox μ 2 ) k Pochhammer SuperscriptBox RowBox ( RowBox - z ) k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 1 0 RowBox 3 1 0 ( RowBox RowBox GridBox RowBox RowBox - ν , RowBox ν + 1 , RowBox RowBox 1 - FractionBox μ 2 ; 1 ; ; RowBox 1 , RowBox RowBox 1 - μ ; RowBox 1 - FractionBox μ 2 ; ; FractionBox 1 2 , RowBox - FractionBox z 2 ) TraditionalForm [/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["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", " ", "z"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]

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

 2001-10-29