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 LegendreP

 http://functions.wolfram.com/07.08.06.0019.01

 Input Form

 LegendreP[\[Nu], \[Mu], 2, z] == ((Gamma[-\[Mu]]/(Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) (z + 1)^(\[Mu]/2) Sum[((Pochhammer[-\[Nu], j] Pochhammer[1 + \[Nu], j] Pochhammer[\[Mu]/2, k - j])/(2^k (Pochhammer[1 + \[Mu], j] j! (k - j)!))) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}])/2^(\[Mu]/2) - (((2^(\[Mu]/2) Sin[Pi \[Nu]] Gamma[\[Mu]])/Pi) Sum[((Pochhammer[-\[Nu] - \[Mu], j] Pochhammer[1 + \[Nu] - \[Mu], j] Pochhammer[\[Mu]/2, k - j])/2^k/(Pochhammer[1 - \[Mu], j] j! (k - j)!)) (z + 1)^k, {k, 0, Infinity}, {j, 0, k}])/ (z + 1)^(\[Mu]/2) /; !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]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "/", "2"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], SuperscriptBox["2", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Mu]", "/", "2"]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", "\[Mu]", "]"]]]], "\[Pi]"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "/", "2"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], SuperscriptBox["2", RowBox[List["-", "k"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]]]]]]]]]]]], "/;", 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 ) 2 - μ 2 Γ ( - μ - ν ) Γ ( ν - μ + 1 ) ( z + 1 ) μ / 2 k = 0 j = 0 k ( μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( z + 1 ) k ( μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] j ! ( k - j ) ! - 2 μ / 2 sin ( π ν ) Γ ( μ ) π ( z + 1 ) - μ 2 k = 0 j = 0 k ( μ 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν - μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] 2 - k ( z + 1 ) k ( 1 - μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], "j"], Pochhammer] j ! ( k - j ) ! /; μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition LegendreP ν μ 2 z Gamma -1 μ 2 -1 μ 2 -1 Gamma -1 μ -1 ν Gamma ν -1 μ 1 -1 z 1 μ 2 -1 j 0 k k 0 Pochhammer μ 2 -1 k -1 j Pochhammer -1 ν j Pochhammer ν 1 j 2 -1 k z 1 k Pochhammer μ 1 j j k -1 j -1 -1 2 μ 2 -1 ν Gamma μ -1 z 1 -1 μ 2 -1 j 0 k k 0 Pochhammer μ 2 -1 k -1 j Pochhammer -1 μ -1 ν j Pochhammer ν -1 μ 1 j 2 -1 k z 1 k Pochhammer 1 -1 μ j j k -1 j -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["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox["2", RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "k"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Mu]", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "k"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], "\[Pi]"]]], "/;", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29