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 JacobiP

 http://functions.wolfram.com/07.15.06.0031.01

 Input Form

 JacobiP[\[Nu], a, b, z] \[Proportional] (Pochhammer[a + b + \[Nu] + 1, \[Nu]]/(2^\[Nu] Gamma[\[Nu] + 1])) z^\[Nu] (1 + O[1/z]) + (((-1)^(a + \[Nu]) 2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] (b + \[Nu])! (a + \[Nu])!)/(Pi (1 + a + b + 2 \[Nu])!)) z^(-1 - a - b - \[Nu]) (1 + O[1/z]) /; (Abs[z] -> Infinity) && Element[a + b + 2 \[Nu] + 1, Integers] && a + b + 2 \[Nu] + 1 >= 0 && Element[a + \[Nu], Integers] && a + \[Nu] >= 0 && b + \[Nu] >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) ( - 1 ) a + ν 2 a + b + ν + 1 sin ( π ν ) ( b + ν ) ! ( a + ν ) ! z - a - b - ν - 1 π ( a + b + 2 ν + 1 ) ! ( 1 + O ( 1 z ) ) + 2 - ν ( a + b + ν + 1 ) ν TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] z ν Γ ( ν + 1 ) ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a + b + 2 ν + 1 a + ν b + ν 0 Condition Proportional JacobiP ν a b z -1 a ν 2 a b ν 1 ν b ν a ν z -1 a -1 b -1 ν -1 a b 2 ν 1 -1 1 O 1 z -1 2 -1 ν Pochhammer a b ν 1 ν z ν Gamma ν 1 -1 1 O 1 z -1 Rule z a b 2 ν 1 a ν b ν 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29