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/07.14.06.0033.01

 Input Form

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

 Standard Form

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

 MathML Form

 C ν λ ( z ) 2 ν Γ ( λ + ν ) Γ ( ν + 1 ) Γ ( λ ) ( z - 1 ) ν k = 0 λ + ν - 1 2 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 1 2 - λ - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( 1 - 2 λ - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k - ( - 1 ) λ + ν + 1 2 2 1 - 2 λ - ν sin ( ν π ) Γ ( λ ) Γ ( λ + ν + 1 ) Γ ( 2 λ + ν ) ( z - 1 ) - 2 λ - ν 2 F 1 ( 2 λ + ν , λ + ν + 1 2 ; 2 λ + 2 ν + 1 ; 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[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], 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] /; λ + ν - 1 2 ν Condition Subscript C ν λ z 2 ν Gamma λ ν Gamma ν 1 Gamma λ -1 z -1 ν k 0 λ ν -1 1 2 Pochhammer -1 ν k Pochhammer 1 2 -1 λ -1 ν k k Pochhammer 1 -1 2 λ -1 2 ν k -1 2 1 -1 z -1 k -1 -1 λ ν 1 2 2 1 -1 2 λ -1 ν ν Gamma λ Gamma λ ν 1 -1 Gamma 2 λ ν z -1 -2 λ -1 ν Hypergeometric2F1 2 λ ν λ ν 1 2 2 λ 2 ν 1 2 1 -1 z -1 λ ν -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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "+", "\[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[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "\[Lambda]", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29