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 LegendreQ

 http://functions.wolfram.com/07.11.03.0027.01

 Input Form

 LegendreQ[n, m, 2, z] == 2^(-1 - n) (-1)^((Abs[m] + m)/2) (1 + z)^n (n + m)! n! ((1 - z)^(Abs[m]/2)/(1 + z)^(Abs[m]/2)) (Sum[(((-1)^(-k + m) (Abs[m] - k - 1)!)/(k! (n - k)! (Abs[m] + n - k)!)) ((z - 1)/(z + 1))^(k - Abs[m]), {k, 0, Abs[m] - 1}] + Sum[(1/(k! (k + Abs[m])! (n - k)! (n - Abs[m] - k)!)) (PolyGamma[1 + k] + PolyGamma[1 + k + Abs[m]] - PolyGamma[n - k + 1] - PolyGamma[n - Abs[m] - k + 1] + Log[1 + z] - Log[1 - z]) ((z - 1)/(z + 1))^k, {k, 0, n - Abs[m]}] + Sum[(((-1)^(n - k + m - 1) (k + Abs[m] - n - 1)!)/ (k! (k + Abs[m])! (n - k)!)) ((z - 1)/(z + 1))^k, {k, n - Abs[m] + 1, n}]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && Abs[m] <= n

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "m", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "m"]], "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "n"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]], RowBox[List["n", "!"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], FractionBox[RowBox[List["Abs", "[", "m", "]"]], "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], FractionBox[RowBox[List["Abs", "[", "m", "]"]], "2"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "+", "m"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], RowBox[List[" ", RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "n", "-", "k"]], ")"]], "!"]]]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], RowBox[List["k", "-", RowBox[List["Abs", "[", "m", "]"]]]]]]]]], " ", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]]]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "-", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "m", "]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "-", "k", "+", "1"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "k"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "+", "1"]]]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "+", "m", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]], "-", "n", "-", "1"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "k"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] n m ( z ) 2 - n - 1 ( - 1 ) m + "\[LeftBracketingBar]" m "\[RightBracketingBar]" 2 ( z + 1 ) n ( m + n ) ! n ! ( 1 - z ) "\[LeftBracketingBar]" m "\[RightBracketingBar]" 2 ( 1 + z ) "\[LeftBracketingBar]" m "\[RightBracketingBar]" 2 ( k = n - "\[LeftBracketingBar]" m "\[RightBracketingBar]" + 1 n ( - 1 ) m + n - k - 1 ( k - n + "\[LeftBracketingBar]" m "\[RightBracketingBar]" - 1 ) ! k ! ( k + "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) ! ( n - k ) ! ( z - 1 z + 1 ) k + k = 0 n - "\[LeftBracketingBar]" m "\[RightBracketingBar]" 1 k ! ( k + "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) ! ( n - k ) ! ( - k + n - "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) ! ( log ( 1 + z ) - log ( 1 - z ) - ψ TagBox["\[Psi]", PolyGamma] ( n - k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( n - "\[LeftBracketingBar]" m "\[RightBracketingBar]" - k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + "\[LeftBracketingBar]" m "\[RightBracketingBar]" + 1 ) ) ( z - 1 z + 1 ) k + k = 0 "\[LeftBracketingBar]" m "\[RightBracketingBar]" - 1 ( - 1 ) m - k ( "\[LeftBracketingBar]" m "\[RightBracketingBar]" - k - 1 ) ! k ! ( n - k ) ! ( n + "\[LeftBracketingBar]" m "\[RightBracketingBar]" - k ) ! ( z - 1 z + 1 ) k - "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) /; n m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] "\[LeftBracketingBar]" m "\[RightBracketingBar]" n Condition LegendreQ n m z 2 -1 n -1 -1 m m 2 -1 z 1 n m n n 1 -1 z m 2 -1 1 z m 2 -1 -1 k n -1 m 1 n -1 m n -1 k -1 k -1 n m -1 k k m n -1 k -1 z -1 z 1 -1 k k 0 n -1 m 1 k k m n -1 k -1 k n -1 m -1 1 z -1 1 -1 z -1 PolyGamma n -1 k 1 -1 PolyGamma n -1 m -1 k 1 PolyGamma k 1 PolyGamma k m 1 z -1 z 1 -1 k k 0 m -1 -1 m -1 k m -1 k -1 k n -1 k n m -1 k -1 z -1 z 1 -1 k -1 m n m m n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["n_", ",", "m_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "m"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "n"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], FractionBox[RowBox[List["Abs", "[", "m", "]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "+", "m"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], RowBox[List["k", "-", RowBox[List["Abs", "[", "m", "]"]]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "n", "-", "k"]], ")"]], "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]]]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "m", "]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "-", "k", "+", "1"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "-", "k"]], ")"]], "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]], "+", "1"]]]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "+", "m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]], "-", "n", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], FractionBox[RowBox[List["Abs", "[", "m", "]"]], "2"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]]]

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

 2001-10-29