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 LegendreQ

 http://functions.wolfram.com/07.12.06.0009.01

 Input Form

 LegendreQ[\[Nu], m, 3, z] == ((((-1)^m Gamma[1 + m + \[Nu]])/(2 m! Gamma[1 - m + \[Nu]])) (((1/2) (Log[z + 1] - Log[z - 1]) - PolyGamma[m] - PolyGamma[1 - m + \[Nu]] - PolyGamma[1 + m + \[Nu]]) (1 - z)^m Hypergeometric2F1[-\[Nu], 1 + \[Nu], m + 1, (1 - z)/2] + ((m! Gamma[1 - m + \[Nu]])/Gamma[1 + m + \[Nu]]) (1 + z)^m Sum[(((m - k - 1)! Pochhammer[-\[Nu], k] Pochhammer[1 + \[Nu], k])/k!) ((z - 1)/2)^k, {k, 0, m - 1}] + ((1 - z^2)^m Sum[((Pochhammer[m - \[Nu], k] Pochhammer[1 + m + \[Nu], k] PolyGamma[k + 1])/(k! Pochhammer[1 + m, k])) ((1 - z)/2)^k, {k, 0, Infinity}])/2^m + (1 - z)^m Sum[((Pochhammer[-\[Nu], k] Pochhammer[1 + \[Nu], k] PolyGamma[1 + k + m])/(k! Pochhammer[1 + m, k])) ((1 - z)/2)^k, {k, 0, Infinity}]))/((z - 1)^(m/2) (z + 1)^(m/2)) /; Abs[(1 - z)/2] < 1 && Element[m, Integers] && m > 0 && ( !Element[\[Nu], Integers] || (Element[\[Nu], Integers] && \[Nu] >= m))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "m", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], "-", RowBox[List["PolyGamma", "[", "m", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], " ", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["m", "!"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "m"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "m"]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν m ( z ) ( - 1 ) m Γ ( m + ν + 1 ) 2 m ! Γ ( ν - m + 1 ) ( z - 1 ) - m / 2 ( z + 1 ) - m / 2 ( ( 1 2 ( log ( z + 1 ) - log ( z - 1 ) ) - ψ TagBox["\[Psi]", PolyGamma] ( m ) - ψ TagBox["\[Psi]", PolyGamma] ( ν - m + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( m + ν + 1 ) ) ( 1 - z ) m 2 F 1 ( - ν , ν + 1 ; m + 1 ; 1 - z 2 ) 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]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( 1 - z ) m k = 0 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + m + 1 ) k ! ( m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "k"], Pochhammer] ( 1 - z 2 ) k + ( z + 1 ) m m ! Γ ( ν - m + 1 ) Γ ( m + ν + 1 ) k = 0 m - 1 ( m - k - 1 ) ! ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z - 1 2 ) k + 2 - m ( 1 - z 2 ) m k = 0 ( m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( m + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) k ! ( m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "k"], Pochhammer] ( 1 - z 2 ) k ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 m + ( ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ν m ) Condition Subscript LegendreQ 𝔔 ν m z -1 m Gamma m ν 1 2 m Gamma ν -1 m 1 -1 z -1 -1 m 2 -1 z 1 -1 m 2 -1 1 2 z 1 -1 z -1 -1 PolyGamma m -1 PolyGamma ν -1 m 1 -1 PolyGamma m ν 1 1 -1 z m Hypergeometric2F1 -1 ν ν 1 m 1 1 -1 z 2 -1 1 -1 z m k 0 Pochhammer -1 ν k Pochhammer ν 1 k PolyGamma k m 1 k Pochhammer m 1 k -1 1 -1 z 2 -1 k z 1 m m Gamma ν -1 m 1 Gamma m ν 1 -1 k 0 m -1 m -1 k -1 Pochhammer -1 ν k Pochhammer ν 1 k k -1 z -1 2 -1 k 2 -1 m 1 -1 z 2 m k 0 Pochhammer m -1 ν k Pochhammer m ν 1 k PolyGamma k 1 k Pochhammer m 1 k -1 1 -1 z 2 -1 k 1 -1 z 2 -1 1 m SuperPlus ν ν ν m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], "-", RowBox[List["PolyGamma", "[", "m", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "2"], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "m"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "||", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "m"]]]], ")"]]]], ")"]]]]]]]]]]

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

 2001-10-29