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 LegendreQ

 http://functions.wolfram.com/07.12.06.0021.01

 Input Form

 LegendreQ[\[Nu], m, 3, z] == (1/(4 m! Gamma[-m - \[Nu]] Gamma[1 - m + \[Nu]])) ((-1)^m (Pi^2 + 2 Pi^2 Cot[Pi \[Nu]]^2 - Log[-1 - z] - Log[1 + z] (1 + 4 Pi Cot[Pi \[Nu]] - 4 Log[1 + z]) - 2 (Pi Cot[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) (PolyGamma[m] + PolyGamma[m - \[Nu]])) ((z + 1)^(m/2)/(z - 1)^(m/2)) Hypergeometric2F1[-\[Nu], 1 + \[Nu], m + 1, (z + 1)/2] + ((2 m! Gamma[-m - \[Nu]])/Gamma[m - \[Nu]]) (Pi Cot[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) ((z - 1)^(m/2)/(z + 1)^(m/2)) Sum[(((m - k - 1)! Pochhammer[-\[Nu], k] Pochhammer[1 + \[Nu], k])/k!) (-((z + 1)/2))^k, {k, 0, m - 1}] + 2^(1 - m) (Pi Cos[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) (z - 1)^(m/2) (z + 1)^(m/2) Sum[((Pochhammer[1 + m, k] Pochhammer[m - \[Nu], k] PolyGamma[k + 1])/(k! Pochhammer[m + \[Nu] + 1, k])) ((z + 1)/2)^k, {k, 0, Infinity}] + (-1)^m 2 (Pi Cos[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) ((z + 1)^(m/2)/(z - 1)^(m/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[1 + \[Nu], k] PolyGamma[k + m + 1])/(k! Pochhammer[m + 1, k])) ((z + 1)/2)^k, {k, 0, Infinity}]) /; Abs[(z + 1)/2] < 1 && Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν m ( z ) 1 4 m ! Γ ( - m - ν ) Γ ( ν - m + 1 ) ( 2 m ! Γ ( - m - ν ) Γ ( m - ν ) ( π cot ( π ν ) + log ( - z - 1 ) - log ( z + 1 ) ) ( z - 1 ) m / 2 ( z + 1 ) m / 2 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 + ( - 1 ) m ( 2 π 2 cot 2 ( π ν ) + π 2 - log ( - z - 1 ) - log ( z + 1 ) ( 4 π cot ( π ν ) - 4 log ( z + 1 ) + 1 ) - 2 ( π cot ( π ν ) + log ( - z - 1 ) - log ( z + 1 ) ) ( ψ TagBox["\[Psi]", PolyGamma] ( m ) + ψ TagBox["\[Psi]", PolyGamma] ( m - ν ) ) ) ( z + 1 ) m / 2 ( z - 1 ) m / 2 2 F 1 ( - ν , ν + 1 ; m + 1 ; z + 1 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["z", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 ( - 1 ) m ( π cos ( π ν ) + log ( - z - 1 ) - log ( z + 1 ) ) ( z + 1 ) m / 2 ( z - 1 ) m / 2 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] ( z + 1 2 ) k + ( - 1 ) m 2 1 - m ( π cos ( π ν ) + log ( - z - 1 ) - log ( z + 1 ) ) ( z - 1 ) m / 2 ( z + 1 ) m / 2 k = 0 ( m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "k"], Pochhammer] ( m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) k ! ( m + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( z + 1 2 ) k ) /; "\[LeftBracketingBar]" z + 1 2 "\[RightBracketingBar]" < 1 m + Condition Subscript LegendreQ 𝔔 ν m z 1 4 m Gamma -1 m -1 ν Gamma ν -1 m 1 -1 2 m Gamma -1 m -1 ν Gamma m -1 ν -1 ν -1 z -1 -1 z 1 z -1 m 2 -1 z 1 m 2 -1 -1 k 0 m -1 m -1 k -1 Pochhammer -1 ν k Pochhammer ν 1 k k -1 -1 z 1 2 -1 k -1 m 2 2 ν 2 2 -1 -1 z -1 -1 z 1 4 ν -1 4 z 1 1 -1 2 ν -1 z -1 -1 z 1 PolyGamma m PolyGamma m -1 ν z 1 m 2 -1 z -1 m 2 -1 -1 Hypergeometric2F1 -1 ν ν 1 m 1 z 1 2 -1 2 -1 m ν -1 z -1 -1 z 1 z 1 m 2 -1 z -1 m 2 -1 -1 k 0 Pochhammer -1 ν k Pochhammer ν 1 k PolyGamma k m 1 k Pochhammer m 1 k -1 z 1 2 -1 k -1 m 2 1 -1 m ν -1 z -1 -1 z 1 z -1 m 2 -1 z 1 m 2 -1 k 0 Pochhammer m 1 k Pochhammer m -1 ν k PolyGamma k 1 k Pochhammer m ν 1 k -1 z 1 2 -1 k z 1 2 -1 1 m SuperPlus [/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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], "]"]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29