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 JacobiP

 http://functions.wolfram.com/07.15.06.0013.01

 Input Form

 JacobiP[\[Nu], a, b, z] == (Gamma[-b]/(Gamma[\[Nu] + 1] Gamma[-b - \[Nu]])) (1 - ((\[Nu] (1 + a + b + \[Nu]))/(2 (1 + b))) (z + 1) - (((1 - \[Nu]) \[Nu] (1 + a + b + \[Nu]) (2 + a + b + \[Nu]))/ (8 (1 + b) (2 + b))) (z + 1)^2 - \[Ellipsis]) - (((Sin[\[Nu] Pi] Gamma[b] Gamma[a + \[Nu] + 1])/ (Pi Gamma[a + b + \[Nu] + 1])) (1 + (((-b - \[Nu]) (1 + a + \[Nu]))/(2 (1 - b))) (z + 1) + (((-b - \[Nu]) (1 - b - \[Nu]) (1 + a + \[Nu]) (2 + a + \[Nu]))/ (8 (1 - b) (2 - b))) (z + 1)^2 + \[Ellipsis]))/((z + 1)/2)^b /; Abs[(z + 1)/2] && !Element[b, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["-", "b"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], " ", "-", "\[Ellipsis]"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]]]]], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "b", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "\[Nu]"]], ")"]]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["b", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) Γ ( - b ) Γ ( ν + 1 ) Γ ( - b - ν ) ( 1 - ν ( a + b + ν + 1 ) 2 ( b + 1 ) ( z + 1 ) - ( 1 - ν ) ν ( a + b + ν + 1 ) ( a + b + ν + 2 ) 8 ( b + 1 ) ( b + 2 ) ( z + 1 ) 2 - ) - sin ( ν π ) Γ ( b ) Γ ( a + ν + 1 ) π Γ ( a + b + ν + 1 ) ( z + 1 2 ) - b ( 1 + ( - b - ν ) ( a + ν + 1 ) 2 ( 1 - b ) ( z + 1 ) + ( - b - ν ) ( - b - ν + 1 ) ( a + ν + 1 ) ( a + ν + 2 ) 8 ( 1 - b ) ( 2 - b ) ( z + 1 ) 2 + ) /; "\[LeftBracketingBar]" z + 1 2 "\[RightBracketingBar]" b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition JacobiP ν a b z Gamma -1 b Gamma ν 1 Gamma -1 b -1 ν -1 1 -1 ν a b ν 1 2 b 1 -1 z 1 -1 1 -1 ν ν a b ν 1 a b ν 2 8 b 1 b 2 -1 z 1 2 -1 -1 ν Gamma b Gamma a ν 1 Gamma a b ν 1 -1 z 1 2 -1 -1 b 1 -1 b -1 ν a ν 1 2 1 -1 b -1 z 1 -1 b -1 ν -1 b -1 ν 1 a ν 1 a ν 2 8 1 -1 b 2 -1 b -1 z 1 2 z 1 2 -1 b [/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["Gamma", "[", RowBox[List["-", "b"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], "-", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "b", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "&&", RowBox[List["!", RowBox[List["b", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29