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 LegendreQ

 http://functions.wolfram.com/07.11.06.0011.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "m", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], FractionBox["m", "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], FractionBox["m", "2"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "m"]], "]"]]]], RowBox[List[RowBox[List["m", "!"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "-", "m"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "2"], ")"]], "k"]]]]]]], " ", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "m"]], "]"]], "/", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "-", "m"]], "]"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]]]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] ν m ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) 1 2 ( 1 + z 1 - z ) - m 2 ( ( z + 1 z - 1 ) m k = 0 m - 1 ( m - k - 1 ) ! ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z - 1 2 ) k + ( - 1 ) m Γ ( m + ν + 1 ) m ! Γ ( ν - m + 1 ) ( log ( 1 + z ) - log ( 1 - z ) - ψ TagBox["\[Psi]", PolyGamma] ( ν - m + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( m + ν + 1 ) ) k = 0 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - z 2 ) k + k = 0 1 k ! ( k + m ) ! ( ( - ν ) k + m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["k", "+", "m"]]], Pochhammer] ( ν + 1 ) k + m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["k", "+", "m"]]], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ( 1 + z 2 ) m + ( - 1 ) m ψ TagBox["\[Psi]", PolyGamma] ( k + m + 1 ) Γ ( m + ν + 1 ) ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] Γ ( ν - m + 1 ) ) ( 1 - z 2 ) k ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 m ν Condition LegendreQ ν m 2 z 1 2 1 z 1 -1 z -1 -1 m 2 -1 z 1 z -1 -1 m k 0 m -1 m -1 k -1 Pochhammer -1 ν k Pochhammer ν 1 k k -1 z -1 2 -1 k -1 m Gamma m ν 1 m Gamma ν -1 m 1 -1 1 z -1 1 -1 z -1 PolyGamma ν -1 m 1 -1 PolyGamma m ν 1 k 0 Pochhammer -1 ν k Pochhammer ν 1 k Pochhammer m 1 k k -1 1 -1 z 2 -1 k k 0 1 k k m -1 Pochhammer -1 ν k m Pochhammer ν 1 k m PolyGamma k 1 1 z 2 -1 m -1 m PolyGamma k m 1 Gamma m ν 1 Pochhammer -1 ν k Pochhammer ν 1 k Gamma ν -1 m 1 -1 1 -1 z 2 -1 k 1 -1 z 2 -1 1 m ν [/itex]

 Rule Form

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

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

 2001-10-29