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 LegendreQ

 http://functions.wolfram.com/07.12.06.0014.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]]]], ")"]]]]]], "/;", " ", 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

 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] n μ ( z ) π csc ( μ π ) 2 π μ ( 1 Γ ( 1 - μ ) ( z + 1 ) μ / 2 ( z - 1 ) μ / 2 k = 0 n ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "k"], Pochhammer] ( 1 - μ ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], "k"], Pochhammer] k ! ( 1 - z 2 ) k - ( n - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] Γ ( μ + 1 ) ( z - 1 ) μ / 2 ( z + 1 ) μ / 2 k = 0 n ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "k"], Pochhammer] ( μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - z 2 ) k ) /; n μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Subscript LegendreQ 𝔔 n μ z μ 2 -1 μ 1 Gamma 1 -1 μ -1 z 1 μ 2 -1 z -1 μ 2 -1 -1 k 0 n Pochhammer -1 n k Pochhammer n 1 k Pochhammer 1 -1 μ k k -1 1 -1 z 2 -1 k -1 Pochhammer n -1 μ 1 2 μ Gamma μ 1 -1 z -1 μ 2 -1 z 1 μ 2 -1 -1 k 0 n Pochhammer -1 n k Pochhammer n 1 k Pochhammer μ 1 k k -1 1 -1 z 2 -1 k n μ [/itex]

 Rule Form

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

 2001-10-29