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 LegendreQ

 http://functions.wolfram.com/07.11.06.0007.01

 Input Form

 LegendreQ[\[Nu], \[Mu], 2, z] == ((Pi Csc[\[Mu] Pi])/2) ((Cos[\[Mu] Pi] 2^(\[Mu]/2) Sum[(((-1)^k Pochhammer[-(\[Mu]/2), k - j] Pochhammer[-\[Nu], j] Pochhammer[\[Nu] + 1, j])/ (Gamma[1 - \[Mu] + j] (k - j)! j! 2^k)) (z - 1)^k, {k, 0, Infinity}, {j, 0, k}])/(1 - z)^(\[Mu]/2) - (Pochhammer[1 + \[Nu] - \[Mu], 2 \[Mu]] (1 - z)^(\[Mu]/2) Sum[(((-1)^k Pochhammer[\[Mu]/2, k - j] Pochhammer[-\[Nu], j] Pochhammer[\[Nu] + 1, j])/(Gamma[1 + \[Mu] + j] (k - j)! j! 2^k)) (z - 1)^k, {k, 0, Infinity}, {j, 0, k}])/2^(\[Mu]/2)) /; Abs[(1 - z)/2] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]]]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", SuperscriptBox["2", RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "j"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], SuperscriptBox["2", "k"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "j"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], SuperscriptBox["2", "k"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] ν μ ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) π csc ( μ π ) 2 ( cos ( μ π ) ( 1 - z ) - μ / 2 2 μ / 2 k = 0 j = 0 k ( - 1 ) 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] ( z - 1 ) k Γ ( j - μ + 1 ) ( k - j ) ! j ! 2 k - ( ν - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] ( 1 - z ) μ 2 2 - μ 2 k = 0 j = 0 k ( - 1 ) 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] ( z - 1 ) k Γ ( j + μ + 1 ) ( k - j ) ! j ! 2 k ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 Condition LegendreQ ν μ 2 z μ 2 -1 μ 1 -1 z -1 μ 2 -1 2 μ 2 -1 j 0 k k 0 -1 k Pochhammer -1 μ 2 -1 k -1 j Pochhammer -1 ν j Pochhammer ν 1 j z -1 k Gamma j -1 μ 1 k -1 j j 2 k -1 -1 Pochhammer ν -1 μ 1 2 μ 1 -1 z μ 2 -1 2 -1 μ 2 -1 j 0 k k 0 -1 k Pochhammer μ 2 -1 k -1 j Pochhammer -1 ν j Pochhammer ν 1 j z -1 k Gamma j μ 1 k -1 j j 2 k -1 1 -1 z 2 -1 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", SuperscriptBox["2", RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "j"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", SuperscriptBox["2", "k"]]]]]]]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox["2", RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "j"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", SuperscriptBox["2", "k"]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]]]]]]]]

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

 2001-10-29