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 LegendreQ

 http://functions.wolfram.com/07.11.06.0023.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "m", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["2", " ", RowBox[List["m", "!"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], RowBox[List["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["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", 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["m", "+", "\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "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["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "m", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "m"], " ", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "1"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] ν m ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) - cos ( π ν ) 2 m ! ( 1 - z 2 ) - m 2 ( ( 1 - z ) m m ! 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] ( z + 1 2 ) k + ( z + 1 ) 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] ( z + 1 2 ) k - ψ TagBox["\[Psi]", PolyGamma] ( m + 1 ) ( z + 1 ) m 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] ) /; "\[LeftBracketingBar]" z + 1 2 "\[RightBracketingBar]" < 1 m + Condition LegendreQ ν m 2 z -1 ν 2 m -1 1 -1 z 2 -1 m 2 -1 1 -1 z m m k 0 m -1 m -1 k -1 Pochhammer -1 ν k Pochhammer ν 1 k k -1 -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 z 1 2 -1 k z 1 m 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 PolyGamma m 1 z 1 m Hypergeometric2F1 -1 ν ν 1 m 1 z 1 2 -1 z 1 2 -1 1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "m"], " ", RowBox[List["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["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], ")"]], "k"]]], RowBox[List["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]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "+", "\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "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["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "m", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "+", "1"]], ",", "k"]], "]"]]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "m"], " ", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "1"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["m", "!"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2001-10-29