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 LegendreP

 http://functions.wolfram.com/07.08.06.0017.01

 Input Form

 LegendreP[\[Nu], \[Mu], 2, z] == (Gamma[-\[Mu]]/(Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) ((1 + z)^(\[Mu]/2)/(1 - z)^(\[Mu]/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 1, k])/ (Pochhammer[1 + \[Mu], k] k!)) ((z + 1)/2)^k, {k, 0, Infinity}] - (((2^\[Mu] Sin[Pi \[Nu]] Gamma[\[Mu]])/Pi) Sum[((Pochhammer[1 - \[Mu] + \[Nu], k] Pochhammer[-\[Mu] - \[Nu], k])/ (Pochhammer[1 - \[Mu], k] k!)) ((z + 1)/2)^k, {k, 0, Infinity}])/ (1 - z^2)^(\[Mu]/2) /; Abs[(z + 1)/2] < 1 && !Element[\[Mu], Integers]

 Standard Form

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

 MathML Form

 P TagBox["P", LegendreP] ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) Γ ( - μ ) Γ ( - μ - ν ) Γ ( ν - μ + 1 ) ( 1 + z ) μ / 2 ( 1 - z ) μ / 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] ( μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z + 1 2 ) k - 2 μ sin ( π ν ) Γ ( μ ) π ( 1 - z 2 ) - μ / 2 k = 0 ( ν - μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] ( - μ - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 1 - μ ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], "k"], Pochhammer] k ! ( z + 1 2 ) k /; "\[LeftBracketingBar]" z + 1 2 "\[RightBracketingBar]" < 1 μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition LegendreP ν μ 2 z Gamma -1 μ Gamma -1 μ -1 ν Gamma ν -1 μ 1 -1 1 z μ 2 -1 1 -1 z μ 2 -1 -1 k 0 Pochhammer -1 ν k Pochhammer ν 1 k Pochhammer μ 1 k k -1 z 1 2 -1 k -1 2 μ ν Gamma μ -1 1 -1 z 2 -1 μ 2 -1 k 0 Pochhammer ν -1 μ 1 k Pochhammer -1 μ -1 ν k Pochhammer 1 -1 μ k k -1 z 1 2 -1 k z 1 2 -1 1 μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "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]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Mu]", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29