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 LegendreQ

 http://functions.wolfram.com/07.12.20.0005.01

 Input Form

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

 Standard Form

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"\[Nu]"], "+", FractionBox["r", RowBox[List["1", "+", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", 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

 2 𝔔 TagBox["\[GothicCapitalQ]", LegendreQ] ν μ ( z ) ν 2 π csc ( μ π ) 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 ν i - 2 S TagBox["S", StirlingS1] k ( i ) r = 1 k ( - 1 ) r ( ν + 1 ) r - 2 ( ( r - 1 ) r ν 2 + i 2 ( ν + 1 ) 2 + ( ( 2 r - 1 ) ν - 1 ) i ( ν + 1 ) ) S TagBox["S", StirlingS1] k ( 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 j = 1 k S TagBox["S", StirlingS1] k ( j ) ν j r = 1 k ( - 1 ) r S TagBox["S", StirlingS1] k ( r ) ( ν + 1 ) r ( ( r - 1 ) r ν 2 + j 2 ( ν + 1 ) 2 + j ( ν + 1 ) ( ( 2 r - 1 ) ν - 1 ) ν 2 ( ν + 1 ) 2 + 2 ( j ν + r ν + 1 ) ( ψ TagBox["\[Psi]", PolyGamma] ( μ + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( ν - μ + 1 ) ) ) + ( ( ψ TagBox["\[Psi]", PolyGamma] ( ν - μ + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( μ + ν + 1 ) ) 2 - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( ν - μ + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( μ + ν + 1 ) ) 𝔓 TagBox["\[GothicCapitalP]", LegendreQ] ν - μ ( z ) ) ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 μ Condition ν 2 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 ν i -2 StirlingS1 k i r 1 k -1 r ν 1 r -2 r -1 r ν 2 i 2 ν 1 2 2 r -1 ν -1 i ν 1 StirlingS1 k 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 j 1 k StirlingS1 k j ν j r 1 k -1 r StirlingS1 k r ν 1 r r -1 r ν 2 j 2 ν 1 2 j ν 1 2 r -1 ν -1 ν 2 ν 1 2 -1 2 j ν -1 r ν 1 -1 PolyGamma μ ν 1 -1 PolyGamma ν -1 μ 1 PolyGamma ν -1 μ 1 -1 PolyGamma μ ν 1 2 -1 PolyGamma 1 ν -1 μ 1 PolyGamma 1 μ ν 1 Subscript LegendreQ 𝔓 ν -1 μ z 1 -1 z 2 -1 1 μ [/itex]

 Rule Form

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

 2001-10-29