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 LegendreQ

 http://functions.wolfram.com/07.10.20.0005.01

 Input Form

 D[LegendreQ[\[Nu], z], {\[Nu], m}] == Sum[(1/k!^2) (((1/2) (Log[1 + z] - Log[1 - z]) + PolyGamma[k + 1]) Sum[Binomial[m, j] D[Pochhammer[-\[Nu], k], {\[Nu], j}] D[Pochhammer[1 + \[Nu], k], {\[Nu], m - j}], {j, 0, m}] - Sum[D[Pochhammer[-\[Nu], k], {\[Nu], q}] D[Pochhammer[1 + \[Nu], k], {\[Nu], r}] KroneckerDelta[q + r + s - m] Multinomial[q, r, s] PolyGamma[s, 1 + \[Nu]], {q, 0, m}, {r, 0, m}, {s, 0, m}]) ((1 - z)/2)^k, {k, 0, Infinity}] /; Abs[(1 - z)/2] < 1 && !Element[\[Nu], Integers] && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "m"]], "}"]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[" ", SuperscriptBox[RowBox[List["k", "!"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[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"]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "j"]], "}"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List["m", "-", "j"]]]], "}"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "m"], RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "q"]], "}"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "r"]], "}"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["q", "+", "r", "+", "s", "-", "m"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List["q", ",", "r", ",", "s"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["s", ",", RowBox[List["1", "+", "\[Nu]"]]]], "]"]]]]]]]]]]]], ")"]], 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["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 m Q TagBox["Q", LegendreQ] ν ( z ) ν m k = 0 1 k ! 2 ( ( 1 2 ( log ( 1 + z ) - log ( 1 - z ) ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ) j = 0 m ( m j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ν j m - j ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ν m - j - q = 0 m r = 0 m s = 0 m q ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ν q r ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ν r δ KroneckerDelta q + r + s - m ( q + r + s ; q , r , s ) TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["q", "+", "r", "+", "s"]], ";", "q"]], ",", "r", ",", "s"]], ")"]], Multinomial, Rule[Editable, False]] ψ TagBox["\[Psi]", PolyGamma] ( s ) ( ν + 1 ) ) ( 1 - z 2 ) k /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m Condition ν m LegendreQ ν z k 0 1 k 2 -1 1 2 1 z -1 1 -1 z PolyGamma k 1 j 0 m Binomial m j ν j Pochhammer -1 ν k ν m -1 j Pochhammer ν 1 k -1 s 0 m r 0 m q 0 m ν q Pochhammer -1 ν k ν r Pochhammer ν 1 k KroneckerDelta q r s -1 m Multinomial q r s PolyGamma s ν 1 1 -1 z 2 -1 k 1 -1 z 2 -1 1 ν m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "m_"]], "}"]]]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[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"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "j"]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List["m", "-", "j"]]]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "m"], RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "q"]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "r"]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["q", "+", "r", "+", "s", "-", "m"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List["q", ",", "r", ",", "s"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["s", ",", RowBox[List["1", "+", "\[Nu]"]]]], "]"]]]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29