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 LegendreQ

 http://functions.wolfram.com/07.11.06.0016.01

 Input Form

 LegendreQ[n, \[Mu], 2, z] == (Pi/2) Csc[\[Mu] Pi] ((Cos[\[Mu] Pi]/Gamma[1 - \[Mu]]) ((1 + z)^(\[Mu]/2)/(1 - z)^(\[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]]) ((1 - z)^(\[Mu]/2)/(1 + z)^(\[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

 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]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], 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["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], 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"]]]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] n μ ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) π csc ( μ π ) 2 ( cos ( μ π ) Γ ( 1 - μ ) ( 1 + z ) μ / 2 ( 1 - z ) 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 ) ( 1 - z ) μ / 2 ( 1 + z ) 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 Condition LegendreQ n μ 2 z μ 2 -1 μ Gamma 1 -1 μ -1 1 z μ 2 -1 1 -1 z 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 1 -1 z μ 2 -1 1 z 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 [/itex]

 Rule Form

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

 2001-10-29