html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 GegenbauerC

 http://functions.wolfram.com/07.14.06.0022.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == (-((2^(1/2 - \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])])/ ((1/2 - \[Lambda])! Sqrt[Pi] Gamma[\[Lambda]]))) (z + 1)^(1/2 - \[Lambda]) Log[(1 + z)/2] Hypergeometric2F1[ 1/2 + \[Nu] + \[Lambda], 1/2 - \[Nu] - \[Lambda], 3/2 - \[Lambda], (1 + z)/2] + ((2^(1 - 2 \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])] (-(1/2) - \[Lambda])! Gamma[\[Nu] + 2 \[Lambda]])/ (Sqrt[Pi] Gamma[\[Lambda]] Gamma[\[Nu] + 1])) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 2 \[Lambda], k])/ (k! Pochhammer[1/2 + \[Lambda], k])) ((1 + z)/2)^k, {k, 0, -\[Lambda] - 1/2}] + ((2^(1/2 - \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])])/ (Sqrt[Pi] Gamma[\[Lambda]])) (z + 1)^(1/2 - \[Lambda]) Sum[((Pochhammer[1/2 - \[Nu] - \[Lambda], k] Pochhammer[ 1/2 + \[Nu] + \[Lambda], k])/(k! (1/2 + k - \[Lambda])!)) (PolyGamma[k + 1] + PolyGamma[3/2 + k - \[Lambda]] - PolyGamma[1/2 + k - \[Nu] - \[Lambda]] - PolyGamma[1/2 + k + \[Nu] + \[Lambda]]) ((1 + z)/2)^k, {k, 0, Infinity}] /; Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], "!"]], SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", "z"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " "]], RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", "\[Lambda]", "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " "]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["1", "2"]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Lambda]"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Nu]", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

 MathML Form

 C ν λ ( z ) - 2 1 2 - λ cos ( π ( λ + ν ) ) ( 1 2 - λ ) ! π Γ ( λ ) log ( z + 1 2 ) ( z + 1 ) 1 2 - λ 2 F 1 ( λ + ν + 1 2 , 1 2 - λ - ν ; 3 2 - λ ; 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["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "\[Lambda]"]], 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] + 2 1 2 - λ cos ( π ( λ + ν ) ) π Γ ( λ ) ( z + 1 ) 1 2 - λ k = 0 1 k ! ( k - λ + 1 2 ) ! ( 1 2 - λ - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( λ + ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - λ + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - λ - ν + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + λ + ν + 1 2 ) ) ( z + 1 2 ) k + 2 1 - 2 λ cos ( π ( λ + ν ) ) ( - λ - 1 2 ) ! Γ ( 2 λ + ν ) π Γ ( λ ) Γ ( ν + 1 ) k = 0 - λ - 1 2 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 2 λ + ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( λ + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( z + 1 2 ) k /; - λ - 1 2 ν Condition Subscript C ν λ z -1 2 1 2 -1 λ λ ν 1 2 -1 λ 1 2 Gamma λ -1 z 1 2 -1 z 1 1 2 -1 λ Hypergeometric2F1 λ ν 1 2 1 2 -1 λ -1 ν 3 2 -1 λ z 1 2 -1 2 1 2 -1 λ λ ν 1 2 Gamma λ -1 z 1 1 2 -1 λ k 0 1 k k -1 λ 1 2 -1 Pochhammer 1 2 -1 λ -1 ν k Pochhammer λ ν 1 2 k PolyGamma k 1 PolyGamma k -1 λ 3 2 -1 PolyGamma k -1 λ -1 ν 1 2 -1 PolyGamma k λ ν 1 2 z 1 2 -1 k 2 1 -1 2 λ λ ν -1 λ -1 1 2 Gamma 2 λ ν 1 2 Gamma λ Gamma ν 1 -1 k 0 -1 λ -1 1 2 Pochhammer -1 ν k Pochhammer 2 λ ν k k Pochhammer λ 1 2 k -1 z 1 2 -1 k -1 λ -1 1 2 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", "z"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], "!"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["1", "2"]]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Nu]", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Lambda]"]], ")"]], "!"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]

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

 2001-10-29