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 LegendreQ

 http://functions.wolfram.com/07.12.06.0034.01

 Input Form

 LegendreQ[\[Nu], \[Mu], 3, z] == (Pi/2) Csc[Pi \[Mu]] E^(Pi I \[Mu]) ((Gamma[-\[Mu]]/(2^(\[Mu]/2) (Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]]))) (((1 - z)^(\[Mu]/2) (1 + z)^(\[Mu]/2))/ (z - 1)^(\[Mu]/2)) Sum[((Pochhammer[\[Mu]/2, k - j] Pochhammer[-\[Nu], j] Pochhammer[1 + \[Nu], j])/ (2^k ((k - j)! j! Pochhammer[1 + \[Mu], j]))) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}] - ((2^(\[Mu]/2) Sin[Pi \[Nu]] Gamma[\[Mu]])/Pi) ((1 - z)^(\[Mu]/2)/ ((z - 1)^(\[Mu]/2) (1 + z)^(\[Mu]/2))) Sum[((Pochhammer[\[Mu]/2, k - j] Pochhammer[-\[Mu] - \[Nu], j] Pochhammer[1 - \[Mu] + \[Nu], j])/ (2^k ((k - j)! j! Pochhammer[1 - \[Mu], j]))) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}] - Pochhammer[\[Nu] - \[Mu] + 1, 2 \[Mu]] (((2^(\[Mu]/2) Gamma[\[Mu]])/(Gamma[\[Mu] - \[Nu]] Gamma[1 + \[Mu] + \[Nu]])) ((z - 1)^(\[Mu]/2)/ ((1 - z)^(\[Mu]/2) (1 + z)^(\[Mu]/2))) Sum[((Pochhammer[-(\[Mu]/2), k - j] Pochhammer[-\[Nu], j] Pochhammer[1 + \[Nu], j])/(2^k ((k - j)! j! Pochhammer[1 - \[Mu], j]))) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}] - ((Sin[Pi \[Nu]] Gamma[-\[Mu]])/(2^(\[Mu]/2) Pi)) (((z - 1)^(\[Mu]/2) (1 + z)^(\[Mu]/2))/(1 - z)^(\[Mu]/2)) Sum[((Pochhammer[-(\[Mu]/2), k - j] Pochhammer[\[Mu] - \[Nu], j] Pochhammer[1 + \[Mu] + \[Nu], j])/(2^k ((k - j)! j! Pochhammer[1 + \[Mu], j]))) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}])) /; !Element[\[Mu], Integers]

 Standard Form

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 MathML Form

 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν μ ( z TagBox["z", HoldComplete[LegendreQ, 3]] ) π 2 csc ( μ π ) π μ ( 2 - μ 2 Γ ( - μ ) Γ ( - μ - ν ) Γ ( ν - μ + 1 ) ( 1 - z ) μ / 2 ( 1 + z ) μ / 2 ( z - 1 ) μ / 2 k = 0 j = 0 k ( μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( k - j ) ! j ! ( μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] ( z + 1 ) k - 2 μ 2 sin ( ν π ) Γ ( μ ) π ( 1 - z ) μ / 2 ( z - 1 ) μ / 2 ( 1 + z ) μ / 2 k = 0 j = 0 k ( μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( - μ + ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( k - j ) ! j ! ( 1 - μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], "j"], Pochhammer] ( z + 1 ) k - ( - μ + ν + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] ( 2 μ / 2 Γ ( μ ) Γ ( μ - ν ) Γ ( μ + ν + 1 ) ( z - 1 ) μ / 2 ( 1 - z ) μ / 2 ( 1 + z ) μ / 2 k = 0 j = 0 k ( - μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( k - j ) ! j ! ( 1 - μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], "j"], Pochhammer] ( z + 1 ) k - 2 - μ 2 sin ( ν π ) Γ ( - μ ) π ( z - 1 ) μ / 2 ( 1 + z ) μ / 2 ( 1 - z ) μ / 2 k = 0 j = 0 k ( - μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( μ + ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( k - j ) ! j ! ( μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] ( z + 1 ) k ) ) /; μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Subscript LegendreQ 𝔔 ν μ HoldComplete LegendreQ 3 z 2 -1 μ μ 2 -1 μ 2 -1 Gamma -1 μ Gamma -1 μ -1 ν Gamma ν -1 μ 1 -1 1 -1 z μ 2 -1 1 z μ 2 -1 z -1 μ 2 -1 -1 j 0 k k 0 Pochhammer μ 2 -1 k -1 j Pochhammer -1 ν j Pochhammer ν 1 j 2 -1 k k -1 j j Pochhammer μ 1 j -1 z 1 k -1 2 μ 2 -1 ν Gamma μ -1 1 -1 z μ 2 -1 z -1 μ 2 -1 1 z μ 2 -1 -1 j 0 k k 0 Pochhammer μ 2 -1 k -1 j Pochhammer -1 μ -1 ν j Pochhammer -1 μ ν 1 j 2 -1 k k -1 j j Pochhammer 1 -1 μ j -1 z 1 k -1 Pochhammer -1 μ ν 1 2 μ 2 μ 2 -1 Gamma μ Gamma μ -1 ν Gamma μ ν 1 -1 z -1 μ 2 -1 1 -1 z μ 2 -1 1 z μ 2 -1 -1 j 0 k k 0 Pochhammer -1 μ 2 -1 k -1 j Pochhammer -1 ν j Pochhammer ν 1 j 2 -1 k k -1 j j Pochhammer 1 -1 μ j -1 z 1 k -1 2 -1 μ 2 -1 ν Gamma -1 μ -1 z -1 μ 2 -1 1 z μ 2 -1 1 -1 z μ 2 -1 -1 j 0 k k 0 Pochhammer -1 μ 2 -1 k -1 j Pochhammer μ -1 ν j Pochhammer μ ν 1 j 2 -1 k k -1 j j Pochhammer μ 1 j -1 z 1 k μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[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"]], "\[Infinity]"], 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18