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 LegendreQ

 http://functions.wolfram.com/07.12.20.0002.01

 Input Form

 D[LegendreQ[\[Nu], \[Mu], 3, z], \[Nu]] == (Pi/2) Csc[Pi \[Mu]] E^(Pi I \[Mu]) (((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) Sum[(1/(Gamma[1 - \[Mu] + k] k!)) ((1 - z)/2)^k Sum[StirlingS1[k, i] \[Nu]^i Sum[(-1)^r StirlingS1[k, r] (r/(1 + \[Nu]) + i/\[Nu]) (1 + \[Nu])^r, {r, 1, k}], {i, 1, k}], {k, 0, Infinity}] - Pochhammer[1 - \[Mu] + \[Nu], 2 \[Mu]] (PolyGamma[1 + \[Mu] + \[Nu]] - PolyGamma[1 - \[Mu] + \[Nu]]) LegendreP[\[Nu], -\[Mu], 3, z] - Pochhammer[\[Nu] - \[Mu] + 1, 2 \[Mu]] ((z - 1)^(\[Mu]/2)/(z + 1)^(\[Mu]/2)) Sum[(1/(Gamma[1 + \[Mu] + k] k!)) ((1 - z)/2)^k Sum[StirlingS1[k, i] \[Nu]^i Sum[(-1)^r StirlingS1[k, r] (r/(1 + \[Nu]) + i/\[Nu]) (1 + \[Nu])^r, {r, 1, k}], {i, 1, k}], {k, 0, Infinity}]) /; Abs[(1 - z)/2] < 1 && !Element[\[Mu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "\[Nu]"], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[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"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", SuperscriptBox["\[Nu]", "i"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["r", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], RowBox[List["-", "1"]]]]], "+", RowBox[List["i", " ", SuperscriptBox["\[Nu]", RowBox[List["-", "1"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "r"]]]]]]]]]]]]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[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"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "k"]], "]"]], RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", SuperscriptBox["\[Nu]", "i"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["r", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], RowBox[List["-", "1"]]]]], "+", RowBox[List["i", " ", SuperscriptBox["\[Nu]", RowBox[List["-", "1"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "r"]]]]]]]]]]]]]]]]], ")"]]]]]], "/;", " ", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν μ ( z ) ν π csc ( π μ ) 2 π μ ( ( z + 1 ) μ / 2 ( z - 1 ) μ / 2 k = 0 1 Γ ( k - μ + 1 ) k ! ( 1 - z 2 ) k i = 1 k S TagBox["S", StirlingS1] k ( i ) ν i r = 1 k ( - 1 ) r S TagBox["S", StirlingS1] k ( r ) ( i ν + r ν + 1 ) ( ν + 1 ) r - ( ν - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] ( z - 1 ) μ / 2 ( z + 1 ) μ / 2 k = 0 1 Γ ( k + μ + 1 ) k ! ( 1 - z 2 ) k i = 1 k S TagBox["S", StirlingS1] k ( i ) ν i r = 1 k ( - 1 ) r S TagBox["S", StirlingS1] k ( r ) ( i ν + r ν + 1 ) ( ν + 1 ) r - ( ν - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( μ + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( ν - μ + 1 ) ) 𝔓 TagBox["\[GothicCapitalP]", LegendreQ] ν - μ ( z ) ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 μ Condition ν Subscript LegendreQ 𝔔 ν μ z μ 2 -1 μ z 1 μ 2 -1 z -1 μ 2 -1 -1 k 0 1 Gamma k -1 μ 1 k -1 1 -1 z 2 -1 k i 1 k StirlingS1 k i ν i r 1 k -1 r StirlingS1 k r i ν -1 r ν 1 -1 ν 1 r -1 Pochhammer ν -1 μ 1 2 μ z -1 μ 2 -1 z 1 μ 2 -1 -1 k 0 1 Gamma k μ 1 k -1 1 -1 z 2 -1 k i 1 k StirlingS1 k i ν i r 1 k -1 r StirlingS1 k r i ν -1 r ν 1 -1 ν 1 r -1 Pochhammer ν -1 μ 1 2 μ PolyGamma μ ν 1 -1 PolyGamma ν -1 μ 1 Subscript LegendreQ 𝔓 ν -1 μ z 1 -1 z 2 -1 1 μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[Nu]_"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", SuperscriptBox["\[Nu]", "i"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["r", RowBox[List["1", "+", "\[Nu]"]]], "+", FractionBox["i", "\[Nu]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "r"]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", SuperscriptBox["\[Nu]", "i"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["r", RowBox[List["1", "+", "\[Nu]"]]], "+", FractionBox["i", "\[Nu]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "r"]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29