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 LegendreP

 http://functions.wolfram.com/07.09.20.0014.01

 Input Form

 D[LegendreP[\[Nu], \[Mu], 3, z], {z, m}] == ((Gamma[1 - \[Mu]/2] Gamma[1 + \[Mu] + \[Nu]])/Gamma[1 - \[Mu] + \[Nu]]) Sum[((2^(2 j - k) Binomial[m, k] k! Gamma[1 - k + m - \[Mu] + \[Nu]])/ ((k - j)! (2 j - k)! Gamma[1 - j - \[Mu]/2] Gamma[1 + k - m + \[Mu] + \[Nu]])) z^(2 j - k) (-1 + z)^((1/2) (-2 j + k - m)) (1 + z)^((1/2) (-2 j + k - m)) LegendreP[\[Nu], k - m + \[Mu], 3, z], {k, 0, m}, {j, 0, k}] /; Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "m"]], "}"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "m", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "j", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "-", "m", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k", "-", "m"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k", "-", "m"]], ")"]]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["k", "-", "m", "+", "\[Mu]"]], ",", "3", ",", "z"]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 m 𝔓 TagBox["\[GothicCapitalP]", LegendreQ] ν μ ( z ) z m Γ ( 1 - μ 2 ) Γ ( μ + ν + 1 ) Γ ( 1 - μ + ν ) k = 0 m j = 0 k 2 2 j - k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k ! Γ ( 1 - k + m - μ + ν ) ( k - j ) ! ( 2 j - k ) ! Γ ( 1 - j - μ 2 ) Γ ( k - m + μ + ν + 1 ) z 2 j - k ( z - 1 ) 1 2 ( - 2 j + k - m ) ( z + 1 ) 1 2 ( - 2 j + k - m ) 𝔓 TagBox["\[GothicCapitalP]", LegendreP] ν k - m + μ ( z TagBox["z", HoldComplete[LegendreP, 3]] ) /; m Condition z m Subscript LegendreQ 𝔓 ν μ z Gamma 1 -1 μ 2 -1 Gamma μ ν 1 Gamma 1 -1 μ ν -1 j 0 k k 0 m 2 2 j -1 k Binomial m k k Gamma 1 -1 k m -1 μ ν k -1 j 2 j -1 k Gamma 1 -1 j -1 μ 2 -1 Gamma k -1 m μ ν 1 -1 z 2 j -1 k z -1 1 2 -2 j k -1 m z 1 1 2 -2 j k -1 m Subscript LegendreP 𝔓 ν k -1 m μ HoldComplete LegendreP 3 z m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "m_"]], "}"]]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "m", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k", "-", "m"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k", "-", "m"]], ")"]]]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["k", "-", "m", "+", "\[Mu]"]], ",", "3", ",", "z"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "j", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "-", "m", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02