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 ChebyshevU

 http://functions.wolfram.com/07.05.06.0044.01

 Input Form

 ChebyshevU[\[Nu], z] == (Sin[2 Pi \[Nu]]/(4 Sqrt[Pi])) Sum[(1/k!) ((Gamma[k - \[Nu]] Gamma[2 + k + \[Nu]] ((-2 I Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)])/ E^((1/2) I Pi Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) - 1/((1/(1 + Subscript[z, 0]))^((1/2) Floor[Arg[z - Subscript[z, 0]]/ (2 Pi)]) (1 + Subscript[z, 0])^((1/2) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]))) Hypergeometric2F1Regularized[k - \[Nu], 2 + k + \[Nu], 3/2 + k, (1 + Subscript[z, 0])/2])/2^k - (Pi Sec[Pi \[Nu]] Sqrt[2] (1 + Subscript[z, 0])^(-2^(-1) - k) Hypergeometric2F1Regularized[3/2 + \[Nu], -(1/2) - \[Nu], 1/2 - k, (1 + Subscript[z, 0])/2])/((1/(1 + Subscript[z, 0]))^ ((1/2) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (1 + Subscript[z, 0])^((1/2) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]))) (z - Subscript[z, 0])^k, {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["4", " ", SqrtBox["\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["z", "0"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], ")"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "k", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox[RowBox[List["1", "+", SubscriptBox["z", "0"]]], "2"]]], "]"]]]], " ", "-", RowBox[List["\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], SqrtBox["2"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "1"]], "/", "2"]], "-", "k"]]], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["z", "0"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox[RowBox[List["1", "+", SubscriptBox["z", "0"]]], "2"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

 U ν ( z ) sin ( 2 π ν ) 4 π k = 0 1 k ! ( 2 - k Γ ( k - ν ) Γ ( k + ν + 2 ) ( - ( 1 z 0 + 1 ) - 1 2 arg ( z - z 0 ) 2 π ( z 0 + 1 ) - 1 2 arg ( z - z 0 ) 2 π - 2 - 1 2 π arg ( z - z 0 ) 2 π arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ) 2 F ~ 1 ( k - ν , k + ν + 2 ; k + 3 2 ; z 0 + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", "\[Nu]", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox["3", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - π sec ( π ν ) 2 ( z 0 + 1 ) - k - 1 2 ( 1 z 0 + 1 ) - 1 2 arg ( z - z 0 ) 2 π ( z 0 + 1 ) - 1 2 arg ( z - z 0 ) 2 π 2 F ~ 1 ( ν + 3 2 , - ν - 1 2 ; 1 2 - k ; z 0 + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "k"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - z 0 ) k ChebyshevU ν z 2 ν 4 1 2 -1 k 0 1 k -1 2 -1 k Gamma k -1 ν Gamma k ν 2 -1 1 Subscript z 0 1 -1 -1 1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 -1 1 2 z -1 Subscript z 0 2 -1 -1 2 -1 1 2 z -1 Subscript z 0 2 -1 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 Hypergeometric2F1Regularized k -1 ν k ν 2 k 3 2 Subscript z 0 1 2 -1 -1 ν 2 1 2 Subscript z 0 1 -1 k -1 1 2 1 Subscript z 0 1 -1 -1 1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 -1 1 2 z -1 Subscript z 0 2 -1 Hypergeometric2F1Regularized ν 3 2 -1 ν -1 1 2 1 2 -1 k Subscript z 0 1 2 -1 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "k", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]]]]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List["4", " ", SqrtBox["\[Pi]"]]]]]]]]

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

 2007-05-02