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 JacobiP

 http://functions.wolfram.com/07.15.20.0001.01

 Input Form

 D[JacobiP[\[Nu], a, b, z], \[Nu]] == (Gamma[1 + a + \[Nu]]/Gamma[\[Nu] + 1]) (PolyGamma[1 + \[Nu] + a] - PolyGamma[1 + \[Nu]]) Hypergeometric2F1Regularized[-\[Nu], a + b + \[Nu] + 1, a + 1, (1 - z)/2] + (((z - 1) Gamma[1 + a + \[Nu]])/(2 Gamma[2 + a] Gamma[1 + \[Nu]])) ((1 + a + b + \[Nu]) HypergeometricPFQ[{{1 - \[Nu], 2 + a + b + \[Nu]}, {1}, {1, -\[Nu]}}, {{2, 2 + a}, {}, {1 - \[Nu]}}, (1 - z)/2, (1 - z)/2] + \[Nu] HypergeometricPFQ[{{1 - \[Nu], 2 + a + b + \[Nu]}, {1}, {1, 1 + a + b + \[Nu]}}, {{2, 2 + a}, {}, {2 + a + b + \[Nu]}}, (1 - z)/2, (1 - z)/2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "\[Nu]"], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "a"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["a", "+", "1"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", RowBox[List["\[Nu]", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) ν ( z - 1 ) Γ ( a + ν + 1 ) 2 Γ ( a + 2 ) Γ ( ν + 1 ) ( ( a + b + ν + 1 ) F ~ 2 0 1 2 1 2 ( 1 - ν , a + b + ν + 2 ; 1 ; 1 , - ν ; 2 , a + 2 ; ; 1 - ν ; 1 - z 2 , 1 - z 2 ) + ν F ~ 2 0 1 2 1 2 ( 1 - ν , a + b + ν + 2 ; 1 ; 1 , a + b + ν + 1 ; 2 , a + 2 ; ; a + b + ν + 2 ; 1 - z 2 , 1 - z 2 ) ) + Γ ( a + ν + 1 ) Γ ( ν + 1 ) ( ψ TagBox["\[Psi]", PolyGamma] ( a + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) ) 2 F ~ 1 ( - ν , a + b + ν + 1 ; a + 1 ; 1 - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] FormBox RowBox FractionBox RowBox RowBox SubsuperscriptBox P ν RowBox ( RowBox a , b ) ( z ) RowBox ν RowBox RowBox FractionBox RowBox RowBox ( RowBox z - 1 ) RowBox Γ ( RowBox a + ν + 1 ) RowBox 2 RowBox Γ ( RowBox a + 2 ) RowBox Γ ( RowBox ν + 1 ) RowBox ( RowBox RowBox RowBox ( RowBox a + b + ν + 1 ) RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 0 1 RowBox 2 1 2 [ RowBox GridBox ErrorBox RowBox RowBox 1 - ν , RowBox RowBox a + b + ν + 2 ; 1 ; 1 , RowBox RowBox - ν ; RowBox 2 , RowBox RowBox RowBox a + 2 ; ; RowBox 1 - ν ; FractionBox RowBox 1 - z 2 , FractionBox RowBox 1 - z 2 ] + RowBox ν RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 0 1 RowBox 2 1 2 ( RowBox RowBox GridBox RowBox RowBox 1 - ν , RowBox RowBox a + b + ν + 2 ; 1 ; 1 , RowBox RowBox a + b + ν + 1 ; RowBox 2 , RowBox RowBox RowBox a + 2 ; ; RowBox a + b + ν + 2 ; FractionBox RowBox 1 - z 2 , FractionBox RowBox 1 - z 2 ) ) + RowBox FractionBox RowBox Γ ( RowBox a + ν + 1 ) RowBox Γ ( RowBox ν + 1 ) RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox a + ν + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox ν + 1 ) ) TagBox TagBox RowBox RowBox SubscriptBox FormBox 2 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox - ν Hypergeometric2F1Regularized Rule Editable , TagBox RowBox a + b + ν + 1 Hypergeometric2F1Regularized Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1Regularized Rule Editable ; TagBox TagBox TagBox RowBox a + 1 Hypergeometric2F1Regularized Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1Regularized Rule Editable ; TagBox FractionBox RowBox 1 - z 2 Hypergeometric2F1Regularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1Regularized TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[Nu]_"]]], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "a"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["a", "+", "1"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", RowBox[List["\[Nu]", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]]]]]]

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

 2001-10-29