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 JacobiP

 http://functions.wolfram.com/07.15.06.0037.01

 Input Form

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

 Standard Form

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 MathML Form

 P ν ( a , b ) ( z ) ( - 1 ) a + b + 2 ν 2 a + 1 sin ( π ν ) Γ ( a + 1 ) π ( a + ν + 1 ) Γ ( 1 - b - ν ) Γ ( a + b + ν + 1 ) ( z - 1 ) - a - 1 3 F 2 ( 1 , 1 , a + 1 ; - b - ν + 1 , a + ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 a + b + ν + 1 sin ( π ν ) Γ ( - a - b - 2 ν - 1 ) Γ ( a + ν + 1 ) π Γ ( - b - ν ) ( z - 1 ) - a - b - ν - 1 k = 0 - a - b - 2 ν - 2 ( b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( a + b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( a + b + 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k + ( - 1 ) a + b + 2 ν - 1 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 + k - 2 ν - 1 ) ! ( log ( z - 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - ν ) - ψ TagBox["\[Psi]", PolyGamma] ( a - k + ν + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - a - b - 2 ν ) ) ( 2 1 - z ) k /; - a - b - 2 ν - 1 + - b - ν + a + ν - 1 z { - 1 , 1 } Condition JacobiP ν a b z -1 a b 2 ν 2 a 1 ν Gamma a 1 a ν 1 Gamma 1 -1 b -1 ν Gamma a b ν 1 -1 z -1 -1 a -1 HypergeometricPFQ 1 1 a 1 -1 b -1 ν 1 a ν 2 2 1 -1 z -1 -1 2 a b ν 1 ν Gamma -1 a -1 b -1 2 ν -1 Gamma a ν 1 Gamma -1 b -1 ν -1 z -1 -1 a -1 b -1 ν -1 k 0 -1 a -1 b -1 2 ν -2 Pochhammer b ν 1 k Pochhammer a b ν 1 k k Pochhammer a b 2 ν 2 k -1 2 1 -1 z -1 k -1 a b 2 ν -1 2 -1 ν z -1 ν Gamma ν 1 Gamma a b ν 1 -1 k 0 a ν Pochhammer -1 a -1 ν k Pochhammer -1 ν k k -1 a -1 b k -1 2 ν -1 -1 z -1 2 -1 PolyGamma k 1 -1 PolyGamma k -1 ν -1 PolyGamma a -1 k ν 1 PolyGamma k -1 a -1 b -1 2 ν 2 1 -1 z -1 k -1 a -1 b -1 2 ν -1 SuperPlus -1 b -1 ν SuperPlus a ν -1 z -1 1 [/itex]

 Rule Form

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

 2001-10-29