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 JacobiP

 http://functions.wolfram.com/07.15.06.0027.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "+", "\[Nu]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]", "-", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "a", "+", "b", "+", "k", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", 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["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " "]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[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["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], ">", "2"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) 2 a + b + ν + 1 Γ ( a + ν + 1 ) Γ ( b + ν + 1 ) sin ( ν π ) sin ( ( a + ν ) π ) π 2 Γ ( a + b + 2 ν + 2 ) log ( z - 1 2 ) ( 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 a + b + ν + 1 sin ( ν π ) Γ ( b + ν + 1 ) π Γ ( - a - ν ) ( z - 1 ) - a - b - ν - 1 k = 0 ( 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 + k + 2 ν + 1 ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b + k + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( - b - k - ν ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b + k + 2 ν + 2 ) ) ( 2 1 - z ) k + 2 - ν ( a + b + ν + 1 ) ν TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] ( z - 1 ) ν Γ ( ν + 1 ) k = 0 a + b + 2 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 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] a + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" > 2 Condition JacobiP ν a b z 2 a b ν 1 Gamma a ν 1 Gamma b ν 1 ν a ν 2 Gamma a b 2 ν 2 -1 log z -1 2 -1 z -1 -1 a -1 b -1 ν -1 Hypergeometric2F1 a b ν 1 b ν 1 a b 2 ν 2 2 1 -1 z -1 -1 2 a b ν 1 ν Gamma b ν 1 Gamma -1 a -1 ν -1 z -1 -1 a -1 b -1 ν -1 k 0 Pochhammer b ν 1 k Pochhammer a b ν 1 k k a b k 2 ν 1 -1 PolyGamma k 1 -1 PolyGamma a b k ν 1 -1 PolyGamma -1 b -1 k -1 ν PolyGamma a b k 2 ν 2 2 1 -1 z -1 k 2 -1 ν Pochhammer a b ν 1 ν z -1 ν Gamma ν 1 -1 k 0 a b 2 ν Pochhammer -1 a -1 ν k Pochhammer -1 ν k k Pochhammer -1 a -1 b -1 2 ν k -1 2 1 -1 z -1 k a b 2 ν 1 a ν 1 -1 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]", "-", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "a", "+", "b", "+", "k", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "k", "+", "1"]], ")"]], "!"]]]]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", 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[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[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["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], ">", "2"]]]]]]]]]]

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

 2001-10-29