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 LegendreQ

 http://functions.wolfram.com/07.11.06.0009.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29