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 JacobiP

 http://functions.wolfram.com/07.15.06.0017.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["b", "-", "1"]]], RowBox[List[RowBox[List["b", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List["Log", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], " ", "-", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]], " "]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], RowBox[List["-", "b"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["b", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "b"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["b", "+", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b", "+", "k", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["b", "\[Element]", "Integers"]], "\[And]", RowBox[List["b", ">", "0"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) - ( b - 1 ) ! sin ( π ν ) Γ ( a + ν + 1 ) π Γ ( a + b + ν + 1 ) ( z + 1 2 ) - b k = 0 b - 1 ( - b - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( a + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], "k"], Pochhammer] ( z + 1 2 ) k + ( - 1 ) b - 1 b ! Γ ( ν + 1 ) Γ ( - b - ν ) log ( z + 1 2 ) 2 F 1 ( - ν , a + b + ν + 1 ; b + 1 ; z + 1 2 ) 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["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["b", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( - 1 ) b Γ ( ν + 1 ) Γ ( - b - ν ) k = 0 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( a + b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( b + k ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( b + k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b + k + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - ν ) ) ( z + 1 2 ) k /; b + Condition JacobiP ν a b z -1 b -1 ν Gamma a ν 1 Gamma a b ν 1 -1 z 1 2 -1 -1 b k 0 b -1 Pochhammer -1 b -1 ν k Pochhammer a ν 1 k k Pochhammer 1 -1 b k -1 z 1 2 -1 k -1 b -1 b Gamma ν 1 Gamma -1 b -1 ν -1 z 1 2 -1 Hypergeometric2F1 -1 ν a b ν 1 b 1 z 1 2 -1 -1 b Gamma ν 1 Gamma -1 b -1 ν -1 k 0 Pochhammer -1 ν k Pochhammer a b ν 1 k k b k -1 PolyGamma k 1 PolyGamma b k 1 -1 PolyGamma a b k ν 1 -1 PolyGamma k -1 ν z 1 2 -1 k b SuperPlus [/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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["b", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["b", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["b", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]]]]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "b"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["b", "+", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b", "+", "k", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "k"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["b", "\[Element]", "Integers"]], "&&", RowBox[List["b", ">", "0"]]]]]]]]]]

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

 2001-10-29