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 LegendreP

 http://functions.wolfram.com/05.07.20.0007.01

 Input Form

 D[LegendreP[n, \[Mu], 2, z], {z, \[Alpha]}] == (\[Mu]/2) z^(1 - \[Alpha]) Sum[(k + 1)! Pochhammer[1 + \[Mu]/2, k] z^k HypergeometricPFQRegularized[{{k + 2}, {-(\[Mu]/2)}, {-n, 1 + n, 1 - \[Mu]/2}}, {{k - \[Alpha] + 2}, {}, {k + 2, 1 - \[Mu]}}, -z, -(z/2)], {k, 0, Infinity}] + (Gamma[1 - \[Mu]/2] Sum[Pochhammer[-(\[Mu]/2), k] (-z)^k HypergeometricPFQRegularized[{{-n, 1 + n, 1 - \[Mu]/2}, {1}, {k + 1}}, {{1, 1 - \[Mu]}, {1 - \[Mu]/2}, {k - \[Alpha] + 1}}, 1/2, -(z/2)], {k, 0, Infinity}])/z^\[Alpha]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", "\[Mu]"]], "2"], SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "2"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]

 MathML Form

 α P TagBox["P", LegendreP] n μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) z α μ 2 z 1 - α k = 0 ( k + 1 ) ! ( μ 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Mu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z k F ~ 1 0 2 1 1 3 ( k + 2 ; - μ 2 ; - n , n + 1 , 1 - μ 2 ; k - α + 2 ; ; k + 2 , 1 - μ ; - z , - z 2 ) + Γ ( 1 - μ 2 ) z - α k = 0 ( - μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "k"], Pochhammer] ( - z ) k F ~ 2 1 1 3 1 1 ( - n , n + 1 , 1 - μ 2 ; 1 ; k + 1 ; 1 , 1 - μ ; 1 - μ 2 ; k - α + 1 ; 1 2 , - z 2 ) FormBox RowBox FractionBox RowBox SuperscriptBox α RowBox SubsuperscriptBox TagBox P LegendreP n μ ( TagBox z HoldComplete LegendreP 2 ) RowBox SuperscriptBox z α RowBox RowBox FractionBox μ 2 SuperscriptBox z RowBox 1 - α RowBox UnderoverscriptBox RowBox k = 0 ErrorBox RowBox RowBox RowBox ( RowBox k + 1 ) ! TagBox SubscriptBox RowBox ( RowBox FractionBox μ 2 + 1 ) k Pochhammer SuperscriptBox z k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 2 RowBox 1 1 3 ( RowBox RowBox GridBox RowBox RowBox RowBox k + 2 ; RowBox - FractionBox μ 2 ; RowBox - n , RowBox n + 1 , RowBox RowBox 1 - FractionBox μ 2 ; RowBox RowBox RowBox RowBox k - α + 2 ; ; RowBox k + 2 , RowBox RowBox 1 - μ ; - z , RowBox - FractionBox z 2 ) + RowBox RowBox Γ ( RowBox 1 - FractionBox μ 2 ) SuperscriptBox z RowBox - α RowBox UnderoverscriptBox RowBox k = 0 RowBox TagBox SubscriptBox RowBox ( RowBox - FractionBox μ 2 ) k Pochhammer SuperscriptBox RowBox ( RowBox - z ) k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 1 1 RowBox 3 1 1 ( RowBox RowBox GridBox RowBox RowBox - n , RowBox n + 1 , RowBox RowBox 1 - FractionBox μ 2 ; 1 ; RowBox k + 1 ; RowBox 1 , RowBox RowBox 1 - μ ; RowBox 1 - FractionBox μ 2 ; RowBox k - α + 1 ; FractionBox 1 2 , RowBox - FractionBox z 2 ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Mu]", " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "2"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]

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

 2001-10-29