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 LegendreP

 http://functions.wolfram.com/07.07.06.0020.01

 Input Form

 LegendreP[\[Nu], z] == (Gamma[\[Nu] + 1]/(2^\[Nu] (Pi^(3/2) Gamma[3/2 + \[Nu]]))) (z - 1)^(-\[Nu] - 1) Sum[(Pochhammer[\[Nu] + 1, k]^2/ (k! Pochhammer[2 \[Nu] + 2, k])) (PolyGamma[k + 1] - PolyGamma[-k - \[Nu]] - PolyGamma[\[Nu] + k + 1] + PolyGamma[2 \[Nu] + k + 2]) (2/(1 - z))^k, {k, 0, Infinity}] - (Sin[Pi \[Nu]]/(2^\[Nu] (Sqrt[Pi] Gamma[-\[Nu]] Gamma[3/2 + \[Nu]]))) (z - 1)^(-\[Nu] - 1) Log[(z - 1)/2] Sum[(Pochhammer[\[Nu] + 1, k]^2/(k! Pochhammer[2 \[Nu] + 2, k])) (2/(1 - z))^k, {k, 0, Infinity}] + (1/(2^\[Nu] Gamma[\[Nu] + 1])) (z - 1)^\[Nu] Sum[(((2 \[Nu] - k)! Pochhammer[-\[Nu], k])/ (k! Gamma[1 - k + \[Nu]])) (2/(1 - z))^k, {k, 0, 2 \[Nu]}] /; Abs[(1 - z)/2] > 1 && Element[\[Nu] + 1/2, Integers] && \[Nu] + 1/2 >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], "2"], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ",", "k"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "k"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "2"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], "2"], " "]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "2"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "\[Nu]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], ">", "1"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 P TagBox["P", LegendreP] ν ( z ) - 2 - ν sin ( π ν ) ( z - 1 ) - ν - 1 π Γ ( - ν ) Γ ( ν + 3 2 ) log ( z - 1 2 ) k = 0 ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] 2 k ! ( 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k + 2 - ν Γ ( ν + 1 ) π 3 / 2 Γ ( ν + 3 2 ) ( z - 1 ) - ν - 1 k = 0 ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] 2 ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + ν + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 2 ν + 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( - k - ν ) ) k ! ( 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k + 2 - ν ( z - 1 ) ν Γ ( ν + 1 ) k = 0 2 ν ( 2 ν - k ) ! ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! Γ ( ν - k + 1 ) ( 2 1 - z ) k /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" > 1 ν + 1 2 Condition LegendreP ν z -1 2 -1 ν ν z -1 -1 ν -1 1 2 Gamma -1 ν Gamma ν 3 2 -1 log z -1 2 -1 k 0 Pochhammer ν 1 k 2 k Pochhammer 2 ν 2 k -1 2 1 -1 z -1 k 2 -1 ν Gamma ν 1 3 2 Gamma ν 3 2 -1 z -1 -1 ν -1 k 0 Pochhammer ν 1 k 2 PolyGamma k 1 -1 PolyGamma k ν 1 PolyGamma k 2 ν 2 -1 PolyGamma -1 k -1 ν k Pochhammer 2 ν 2 k -1 2 1 -1 z -1 k 2 -1 ν z -1 ν Gamma ν 1 -1 k 0 2 ν 2 ν -1 k Pochhammer -1 ν k k Gamma ν -1 k 1 -1 2 1 -1 z -1 k 1 -1 z 2 -1 1 ν 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "k"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "2"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "\[Nu]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "\[Nu]"]], "]"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29