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 LegendreQ

 http://functions.wolfram.com/07.12.20.0012.02

 Input Form

 D[LegendreQ[\[Nu], \[Mu], 3, z], {z, n}] == (Pi/2) Csc[Pi \[Mu]] E^(Pi I \[Mu]) (((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2 + n)) Gamma[1 + \[Mu]/2] Sum[Binomial[n, j] Hypergeometric2F1Regularized[-j, \[Mu]/2, 1 - j + \[Mu]/2, (z + 1)/(z - 1)] HypergeometricPFQRegularized[{1, -\[Nu], 1 + \[Nu]}, {1 - n + j, 1 - \[Mu]}, (1 - z)/2] ((z - 1)/(z + 1))^j, {j, 0, n}] - Pochhammer[1 + \[Nu] - \[Mu], 2 \[Mu]] ((z - 1)^(\[Mu]/2 - n)/ (z + 1)^(\[Mu]/2)) Gamma[1 - \[Mu]/2] Sum[Binomial[n, j] Hypergeometric2F1Regularized[-j, -(\[Mu]/2), 1 - j - \[Mu]/2, (z + 1)/(z - 1)] HypergeometricPFQRegularized[ {1, -\[Nu], 1 + \[Nu]}, {1 - n + j, 1 + \[Mu]}, (1 - z)/2] ((z - 1)/(z + 1))^j, {j, 0, n}]) /; Element[n, Integers] && n >= 0 && !Element[\[Mu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["\[Mu]", "/", "2"]], "+", "n"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", FractionBox["\[Mu]", "2"], ",", RowBox[List["1", "-", "j", "+", FractionBox["\[Mu]", "2"]]], ",", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]]]], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "n", "+", "j"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "j"]]]]]]], " ", "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["\[Mu]", "/", "2"]], "-", "n"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["1", "-", "j", "-", FractionBox["\[Mu]", "2"]]], ",", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]]]], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "n", "+", "j"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "j"]]]]]]]]], " ", ")"]]]]]], " ", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 n 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν μ ( z ) z n π csc ( μ π ) 2 π μ ( Γ ( μ 2 + 1 ) ( z + 1 ) μ / 2 ( z - 1 ) μ / 2 + n j = 0 n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 2 F ~ 1 ( - j , μ 2 ; μ 2 - j + 1 ; z + 1 z - 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "j"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[FractionBox["\[Mu]", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Mu]", "2"], "-", "j", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 3 F ~ 2 ( 1 , - ν , ν + 1 ; j - n + 1 , 1 - μ ; 1 - 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["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "-", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( z - 1 z + 1 ) j - ( ν - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] Γ ( 1 - μ 2 ) ( z - 1 ) μ / 2 - n ( z + 1 ) μ / 2 j = 0 n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 2 F ~ 1 ( - j , - μ 2 ; 1 - j - μ 2 ; z + 1 z - 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "j"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Mu]", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "j", "-", FractionBox["\[Mu]", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 3 F ~ 2 ( 1 , - ν , ν + 1 ; j - n + 1 , μ + 1 ; 1 - 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["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "-", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( z - 1 z + 1 ) j ) /; n μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z n Subscript LegendreQ 𝔔 ν μ z μ 2 -1 μ Gamma μ 2 -1 1 z 1 μ 2 -1 z -1 μ 2 -1 n -1 j 0 n Binomial n j Hypergeometric2F1Regularized -1 j μ 2 -1 μ 2 -1 -1 j 1 z 1 z -1 -1 HypergeometricPFQRegularized 1 -1 ν ν 1 j -1 n 1 1 -1 μ 1 -1 z 2 -1 z -1 z 1 -1 j -1 Pochhammer ν -1 μ 1 2 μ Gamma 1 -1 μ 2 -1 z -1 μ 2 -1 -1 n z 1 μ 2 -1 -1 j 0 n Binomial n j Hypergeometric2F1Regularized -1 j -1 μ 2 -1 1 -1 j -1 μ 2 -1 z 1 z -1 -1 HypergeometricPFQRegularized 1 -1 ν ν 1 j -1 n 1 μ 1 1 -1 z 2 -1 z -1 z 1 -1 j n μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", FractionBox["\[Mu]", "2"], ",", RowBox[List["1", "-", "j", "+", FractionBox["\[Mu]", "2"]]], ",", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "n", "+", "j"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "j"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[FractionBox["\[Mu]", "2"], "+", "n"]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[FractionBox["\[Mu]", "2"], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["1", "-", "j", "-", FractionBox["\[Mu]", "2"]]], ",", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "n", "+", "j"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "j"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29