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 LegendreP

 http://functions.wolfram.com/07.09.06.0002.01

 Input Form

 LegendreP[\[Nu], \[Mu], 3, z] == ((1 - z)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) ((\[Mu]/2) Sum[(Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 1, k] Pochhammer[\[Mu]/2 + 1, m] Pochhammer[1 - \[Mu]/2, k]) Pochhammer[-\[Mu]/2, j] (-1)^(j + k) (z^(j + k + m + 1)/ (Pochhammer[2, m + k] Gamma[1 - \[Mu] + k] k! j! 2^k)), {k, 0, Infinity}, {m, 0, Infinity}, {j, 0, Infinity}] + Sum[(Pochhammer[-\[Nu], k + m] Pochhammer[\[Nu] + 1, k + m] Pochhammer[m - \[Mu]/2 + 1, k] Pochhammer[-\[Mu]/2, j] (-z)^(j + k))/ (Gamma[1 - \[Mu] + k + m] (k + m)! j! k! 2^(k + m)), {k, 0, Infinity}, {m, 0, Infinity}, {j, 0, Infinity}]) /; Abs[z] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", "\[Mu]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], 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"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["\[Mu]", "/", "2"]], "+", "1"]], ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["\[Mu]", "/", "2"]]]], ",", "k"]], "]"]]]], ")"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]], ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], RowBox[List[SuperscriptBox["z", RowBox[List["j", "+", "k", "+", "m", "+", "1"]]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["2", ",", RowBox[List["m", "+", "k"]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]], RowBox[List["j", "!"]], SuperscriptBox["2", "k"]]], ")"]]]]]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["\[Mu]", "/", "2"]], "+", "1"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]], ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["j", "+", "k"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k", "+", "m"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], RowBox[List["k", "!"]], SuperscriptBox["2", RowBox[List["k", "+", "m"]]]]], ")"]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]

 MathML Form

 𝔓 TagBox["\[GothicCapitalP]", LegendreQ] ν μ ( z ) ( 1 - z ) μ / 2 ( z - 1 ) μ / 2 ( μ 2 k = 0 m = 0 j = 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 2 + 1 ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Mu]", "2"], "+", "1"]], ")"]], "m"], Pochhammer] ( 1 - μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ")"]], "k"], Pochhammer] ( - μ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "j"], Pochhammer] ( - 1 ) j + k z j + k + m + 1 ( 2 ) k + m TagBox[SubscriptBox[RowBox[List["(", "2", ")"]], RowBox[List["k", "+", "m"]]], Pochhammer] Γ ( k - μ + 1 ) k ! j ! 2 k + k = 0 m = 0 j = 0 ( - ν ) k + m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["k", "+", "m"]]], Pochhammer] ( ν + 1 ) k + m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["k", "+", "m"]]], Pochhammer] ( m - μ 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", FractionBox["\[Mu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( - μ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "j"], Pochhammer] ( - z ) j + k Γ ( k + m - μ + 1 ) ( k + m ) ! k ! j ! 2 k + m ) /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 Condition Subscript LegendreQ 𝔓 ν μ z 1 -1 z μ 2 -1 z -1 μ 2 -1 -1 μ 2 -1 j 0 m 0 k 0 Pochhammer -1 ν k Pochhammer ν 1 k Pochhammer μ 2 -1 1 m Pochhammer 1 -1 μ 2 -1 k Pochhammer -1 μ 2 -1 j -1 j k z j k m 1 Pochhammer 2 k m Gamma k -1 μ 1 k j 2 k -1 j 0 m 0 k 0 Pochhammer -1 ν k m Pochhammer ν 1 k m Pochhammer m -1 μ 2 -1 1 k Pochhammer -1 μ 2 -1 j -1 z j k Gamma k m -1 μ 1 k m k j 2 k m -1 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Mu]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "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"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["\[Mu]", "2"], "+", "1"]], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["z", RowBox[List["j", "+", "k", "+", "m", "+", "1"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["2", ",", RowBox[List["m", "+", "k"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["j", "!"]], " ", SuperscriptBox["2", "k"]]]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["k", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", FractionBox["\[Mu]", "2"], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["j", "+", "k"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k", "+", "m"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", RowBox[List["k", "+", "m"]]]]]]]]]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]

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

 2001-10-29