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 JacobiP

 http://functions.wolfram.com/07.15.21.0002.01

 Input Form

 Integrate[z^(\[Alpha] - 1) JacobiP[\[Nu], a, b, z], z] == ((Pochhammer[a + 1, \[Nu]] z^\[Alpha])/(\[Alpha] Gamma[\[Nu] + 1])) HypergeometricPFQRegularized[{{-\[Nu], a + b + \[Nu] + 1}, {\[Alpha]}, {}}, {{a + 1}, {\[Alpha] + 1}, {}}, -(z/2), 1/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "1"]], ",", "\[Nu]"]], "]"]], SuperscriptBox["z", "\[Alpha]"]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]]], "}"]], ",", RowBox[List["{", "\[Alpha]", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["\[Alpha]", "+", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "2"]]], ",", FractionBox["1", "2"]]], "]"]]]]]]]]

 MathML Form

 z α - 1 P ν ( a , b ) ( z ) z ( a + 1 ) ν TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] z α α Γ ( ν + 1 ) F ~ 1 1 0 2 1 0 ( - ν , a + b + ν + 1 ; α ; ; a + 1 ; α + 1 ; ; - z 2 , 1 2 ) FormBox RowBox RowBox RowBox RowBox SuperscriptBox z RowBox α - 1 RowBox SubsuperscriptBox P ν RowBox ( RowBox a , b ) ( z ) RowBox z RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox a + 1 ) ν Pochhammer SuperscriptBox z α RowBox α RowBox Γ ( RowBox ν + 1 ) RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 1 0 RowBox 2 1 0 [ RowBox GridBox ErrorBox RowBox RowBox - ν , RowBox RowBox a + b + ν + 1 ; α ; ; RowBox RowBox a + 1 ; RowBox α + 1 ; ; - FractionBox z 2 , FractionBox 1 2 ] TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "1"]], ",", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]]], "}"]], ",", RowBox[List["{", "\[Alpha]", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["\[Alpha]", "+", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "2"]]], ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]]]]]]

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

 2001-10-29