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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0045.01

 Input Form

 ChebyshevT[\[Nu], z] == ((\[Nu] Sin[2 Pi \[Nu]])/(2 Sqrt[Pi])) Sum[(1/k!) ((-2^(-k)) Gamma[k - \[Nu]] Gamma[k + \[Nu]] (-2 I I^Floor[Arg[z - x]/(2 Pi)] Floor[Arg[z - x]/(2 Pi)] + Exp[Pi I Floor[Arg[z - x]/(2 Pi)]]) Hypergeometric2F1Regularized[ k - \[Nu], k + \[Nu], 1/2 + k, (1/2) (1 + x)] + ((Pi Sec[Pi \[Nu]])/Sqrt[2]) (1 + x)^(1/2 - k) Exp[Pi I Floor[Arg[z - x]/(2 Pi)]] Hypergeometric2F1Regularized[ 1/2 + \[Nu], 1/2 - \[Nu], 3/2 - k, (1/2) (1 + x)]) (z - x)^k, {k, 0, Infinity}] /; Element[x, Reals] && x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], 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["k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["k", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]]]]]], "]"]]]], "+", " ", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], SqrtBox["2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k"]]], " ", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "k"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]]]]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 T ν ( z ) ν sin ( 2 π ν ) 2 π k = 0 1 k ! ( π sec ( π ν ) 2 ( x + 1 ) 1 2 - k π arg ( z - x ) 2 π 2 F ~ 1 ( ν + 1 2 , 1 2 - ν ; 3 2 - k ; x + 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["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "k"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 2 - k Γ ( k - ν ) Γ ( k + ν ) ( - 2 arg ( z - x ) 2 π arg ( z - x ) 2 π + π arg ( z - x ) 2 π ) 2 F ~ 1 ( k - ν , k + ν ; k + 1 2 ; x + 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]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition ChebyshevT ν z ν 2 ν 2 1 2 -1 k 0 1 k -1 ν 2 1 2 -1 x 1 1 2 -1 k z -1 x 2 -1 Hypergeometric2F1Regularized ν 1 2 1 2 -1 ν 3 2 -1 k x 1 2 -1 -1 2 -1 k Gamma k -1 ν Gamma k ν -2 z -1 x 2 -1 z -1 x 2 -1 z -1 x 2 -1 Hypergeometric2F1Regularized k -1 ν k ν k 1 2 x 1 2 -1 z -1 x k x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", 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[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "k"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["k", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "k"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], SqrtBox["2"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02