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 GegenbauerC

 http://functions.wolfram.com/05.09.06.0029.01

 Input Form

 GegenbauerC[n, \[Lambda], z] == ((2^n Sqrt[Pi] Gamma[\[Lambda] + n/2])/(Gamma[\[Lambda]] Gamma[(1 - n)/2] n!)) Sum[((Pochhammer[-(n/2), k] Pochhammer[\[Lambda] + n/2, k])/ (Pochhammer[1/2, k] k!)) z^(2 k), {k, 0, Floor[n/2]}] + ((2^n Sqrt[Pi] z Gamma[\[Lambda] + (1 + n)/2])/ (Gamma[\[Lambda]] Gamma[1 - n/2] (n - 1)!)) Sum[((Pochhammer[(1 - n)/2, k] Pochhammer[\[Lambda] + (n + 1)/2, k])/ (Pochhammer[3/2, k] k!)) z^(2 k), {k, 0, Floor[(n - 1)/2]}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", FractionBox["n", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "n"]], "2"], "]"]], " ", RowBox[List["n", "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["n", "2"]]], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["n", "+", "1"]], "2"]]], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]]]]]]]]

 MathML Form

 C n λ ( z ) 2 n π Γ ( λ + n 2 ) Γ ( λ ) Γ ( 1 - n 2 ) n ! k = 0 n 2 ( - n 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["n", "2"]]], ")"]], "k"], Pochhammer] ( λ + n 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", FractionBox["n", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! z 2 k + 2 n π z Γ ( λ + n + 1 2 ) Γ ( λ ) Γ ( 1 - n 2 ) ( n - 1 ) ! k = 0 n - 1 2 ( 1 - n 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "n"]], "2"], ")"]], "k"], Pochhammer] ( n + 1 2 + λ ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], "+", "\[Lambda]"]], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! z 2 k Subscript C n λ z 2 n 1 2 Gamma λ n 2 -1 Gamma λ Gamma 1 -1 n 2 -1 n -1 k 0 n 2 -1 Pochhammer -1 n 2 -1 k Pochhammer λ n 2 -1 k Pochhammer 1 2 k k -1 z 2 k 2 n 1 2 z Gamma λ n 1 2 -1 Gamma λ Gamma 1 -1 n 2 -1 n -1 -1 k 0 n -1 2 -1 Pochhammer 1 -1 n 2 -1 k Pochhammer n 1 2 -1 λ k Pochhammer 3 2 k k -1 z 2 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", FractionBox["n", "2"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["n", "2"]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "n"]], "2"], "]"]], " ", RowBox[List["n", "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["n", "+", "1"]], "2"]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]]]]]]]]

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

 2007-05-02