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 LegendreP

 http://functions.wolfram.com/07.08.06.0022.01

 Input Form

 LegendreP[\[Nu], m, 2, z] == ((-1)^(m - 1)/(m! Gamma[-m - \[Nu]] Gamma[1 - m + \[Nu]])) ((1 + z)^(m/2)/(1 - z)^(m/2)) Log[(z + 1)/2] Hypergeometric2F1[-\[Nu], 1 + \[Nu], 1 + m, (z + 1)/2] - (((2^m Sin[Pi \[Nu]] (m - 1)!)/Pi) Sum[((Pochhammer[-m - \[Nu], k] Pochhammer[1 - m + \[Nu], k])/ (k! Pochhammer[1 - m, k])) ((z + 1)/2)^k, {k, 0, m - 1}])/ (1 - z^2)^(m/2) + ((-1)^m/(Gamma[-m - \[Nu]] Gamma[1 - m + \[Nu]])) ((1 + z)^(m/2)/(1 - z)^(m/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[1 + \[Nu], k])/(k! (k + m)!)) (PolyGamma[k + 1] + PolyGamma[m + k + 1] - PolyGamma[k - \[Nu]] - PolyGamma[\[Nu] + k + 1]) ((z + 1)/2)^k, {k, 0, Infinity}] /; Abs[(z + 1)/2] < 1 && Element[m, Integers] && m > 0 && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "m", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "m"], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]], " "]], "\[Pi]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "k", "+", "1"]], "]"]]]], ")"]], 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["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 P TagBox["P", LegendreP] ν m ( z TagBox["z", HoldComplete[LegendreP, 2]] ) ( - 1 ) m - 1 m ! Γ ( - m - ν ) Γ ( ν - m + 1 ) log ( z + 1 2 ) ( 1 + z ) m / 2 ( 1 - z ) m / 2 2 F 1 ( - ν , ν + 1 ; m + 1 ; z + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 m sin ( π ν ) ( m - 1 ) ! π ( 1 - z 2 ) - m / 2 k = 0 m - 1 ( - m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν - m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "m", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"], Pochhammer] ( z + 1 2 ) k + ( - 1 ) m Γ ( - m - ν ) Γ ( - m + ν + 1 ) ( 1 + z ) m / 2 ( 1 - z ) m / 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] k ! ( k + m ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + m + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - ν ) ) ( z + 1 2 ) k /; "\[LeftBracketingBar]" z + 1 2 "\[RightBracketingBar]" < 1 m + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition LegendreP ν m 2 z -1 m -1 m Gamma -1 m -1 ν Gamma ν -1 m 1 -1 log z 1 2 -1 1 z m 2 -1 1 -1 z m 2 -1 -1 Hypergeometric2F1 -1 ν ν 1 m 1 z 1 2 -1 -1 2 m ν m -1 -1 1 -1 z 2 -1 m 2 -1 k 0 m -1 Pochhammer -1 m -1 ν k Pochhammer ν -1 m 1 k k Pochhammer 1 -1 m k -1 z 1 2 -1 k -1 m Gamma -1 m -1 ν Gamma -1 m ν 1 -1 1 z m 2 -1 1 -1 z m 2 -1 -1 k 0 Pochhammer -1 ν k Pochhammer ν 1 k k k m -1 PolyGamma k 1 PolyGamma k m 1 -1 PolyGamma k ν 1 -1 PolyGamma k -1 ν z 1 2 -1 k z 1 2 -1 1 m SuperPlus ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "2"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "m"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m"]], ",", "k"]], "]"]]]]]]]]], "\[Pi]"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "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["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "k", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29