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 JacobiP

 http://functions.wolfram.com/07.15.06.0030.01

 Input Form

 JacobiP[\[Nu], a, b, z] == (((-1)^(a + \[Nu]) 2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] (b + \[Nu])! (a + \[Nu])!)/(Pi (1 + a + b + 2 \[Nu])!)) (z - 1)^(-1 - a - b - \[Nu]) Hypergeometric2F1[1 + a + b + \[Nu], 1 + b + \[Nu], 2 + a + b + 2 \[Nu], 2/(1 - z)] + ((a + b + 2 \[Nu])!/(2^\[Nu] (Gamma[1 + \[Nu]] Gamma[1 + a + b + \[Nu]]))) (z - 1)^\[Nu] Sum[((Pochhammer[-a - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! Pochhammer[-a - b - 2 \[Nu], k])) (2/(1 - z))^k, {k, 0, a + \[Nu]}] /; Element[1 + a + b + 2 \[Nu], Integers] && 1 + a + b + 2 \[Nu] >= 0 && Element[a + \[Nu], Integers] && a + \[Nu] >= 0 && b + \[Nu] >= 0 && !IntervalMemberQ[Interval[{-1, 1}], z]

 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[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[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "b", "+", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["a", "+", "\[Nu]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) ( - 1 ) a + ν 2 a + b + ν + 1 sin ( π ν ) ( b + ν ) ! ( a + ν ) ! π ( a + b + 2 ν + 1 ) ! ( z - 1 ) - a - b - ν - 1 2 F 1 ( a + b + ν + 1 , b + ν + 1 ; a + b + 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 - ν ( a + b + 2 ν ) ! ( z - 1 ) ν Γ ( ν + 1 ) Γ ( a + b + ν + 1 ) k = 0 a + ν ( ( - a - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( - a - b - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k /; a + b + 2 ν + 1 a + ν b + ν 0 z { - 1 , 1 } FormBox RowBox RowBox RowBox SubsuperscriptBox P ν RowBox ( RowBox a , b ) ( z ) RowBox RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox a + ν SuperscriptBox 2 RowBox a + b + ν + 1 RowBox sin ( RowBox π ν ) RowBox RowBox ( RowBox b + ν ) ! RowBox RowBox ( RowBox a + ν ) ! RowBox π RowBox RowBox ( RowBox a + b + RowBox 2 ν + 1 ) ! SuperscriptBox RowBox ( RowBox z - 1 ) RowBox RowBox - a - b - ν - 1 TagBox TagBox RowBox RowBox SubscriptBox FormBox 2 TraditionalForm SubscriptBox F FormBox 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox a + b + ν + 1 Hypergeometric2F1 Rule Editable , TagBox RowBox b + ν + 1 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox RowBox a + b + RowBox 2 ν + 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox FractionBox 2 RowBox 1 - z Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 + RowBox FractionBox RowBox SuperscriptBox 2 RowBox - ν RowBox RowBox ( RowBox a + b + RowBox 2 ν ) ! SuperscriptBox RowBox ( RowBox z - 1 ) ν RowBox RowBox Γ ( RowBox ν + 1 ) RowBox Γ ( RowBox a + b + ν + 1 ) RowBox UnderoverscriptBox RowBox k = 0 RowBox a + ν ErrorBox RowBox FractionBox RowBox ( RowBox TagBox SubscriptBox RowBox ( RowBox RowBox - a - ν ) k Pochhammer TagBox SubscriptBox RowBox ( RowBox - ν ) k Pochhammer RowBox RowBox k ! TagBox SubscriptBox RowBox ( RowBox RowBox - a - b - RowBox 2 ν ) k Pochhammer SuperscriptBox RowBox ( FractionBox 2 RowBox 1 - z ) k /; RowBox RowBox RowBox a + b + RowBox 2 ν + 1 RowBox RowBox a + ν RowBox RowBox b + ν 0 RowBox z RowBox { RowBox RowBox - 1 , 1 } TraditionalForm [/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[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[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "b", "+", "\[Nu]"]], ",", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["a", "+", "\[Nu]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[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"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]

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

 2001-10-29

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