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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0060.01

 Input Form

 ChebyshevT[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; Subscript[F, m][z, \[Nu]] == Cos[\[Nu] Pi] Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu], k])/ (Pochhammer[1/2, k] k!)) ((z + 1)/2)^k, {k, 0, m}] + Sqrt[2] \[Nu] Sin[\[Nu] Pi] Sqrt[1 + z] Sum[((Pochhammer[1/2 - \[Nu], k] Pochhammer[1/2 + \[Nu], k])/ (Pochhammer[3/2, k] k!)) ((z + 1)/2)^k, {k, 0, m}] == ChebyshevT[\[Nu], z] - ((2^(-1 - m) Cos[Pi \[Nu]] Pochhammer[-\[Nu], 1 + m] Pochhammer[\[Nu], 1 + m])/((1 + m)! Pochhammer[1/2, 1 + m])) (1 + z)^(1 + m) HypergeometricPFQ[{1, 1 + m - \[Nu], 1 + m + \[Nu]}, {3/2 + m, 2 + m}, (1 + z)/2] - ((2^(-(1/2) - m) \[Nu] Sin[Pi \[Nu]] Pochhammer[1/2 - \[Nu], 1 + m] Pochhammer[1/2 + \[Nu], 1 + m])/((1 + m)! Pochhammer[3/2, 1 + m])) (1 + z)^(3/2 + m) HypergeometricPFQ[{1, 3/2 + m - \[Nu], 3/2 + m + \[Nu]}, {2 + m, 5/2 + m}, (1 + z)/2] && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], SqrtBox[RowBox[List["1", "+", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "m"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["1", "+", "m"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "m"]], ",", RowBox[List["2", "+", "m"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], "-", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], ")"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["3", "2"], "+", "m"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List[FractionBox["5", "2"], "+", "m"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]]]]]], " ", StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 T ν ( z ) F ( z , ν ) /; ( ( F m ( z , ν ) cos ( ν π ) k = 0 m ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν ) k TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! ( z + 1 2 ) k + 2 z + 1 ν sin ( ν π ) k = 0 m ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( z + 1 2 ) k T ν ( z ) - 2 - m - 1 cos ( π ν ) ( - ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( m + 1 ) ! ( 1 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( z + 1 ) m + 1 3 F 2 ( 1 , m - ν + 1 , m + ν + 1 ; m + 3 2 , m + 2 ; z + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 - m - 1 2 ν sin ( π ν ) ( 1 2 - ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( ν + 1 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( m + 1 ) ! ( 3 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( z + 1 ) m + 3 2 3 F 2 ( 1 , m - ν + 3 2 , m + ν + 3 2 ; m + 2 , m + 5 2 ; z + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["5", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) m ) Condition ChebyshevT ν z Subscript F z ν Subscript F m z ν ν k 0 m Pochhammer -1 ν k Pochhammer ν k Pochhammer 1 2 k k -1 z 1 2 -1 k 2 1 2 z 1 1 2 ν ν k 0 m Pochhammer 1 2 -1 ν k Pochhammer ν 1 2 k Pochhammer 3 2 k k -1 z 1 2 -1 k ChebyshevT ν z -1 2 -1 m -1 ν Pochhammer -1 ν m 1 Pochhammer ν m 1 m 1 Pochhammer 1 2 m 1 -1 z 1 m 1 HypergeometricPFQ 1 m -1 ν 1 m ν 1 m 3 2 m 2 z 1 2 -1 -1 2 -1 m -1 1 2 ν ν Pochhammer 1 2 -1 ν m 1 Pochhammer ν 1 2 m 1 m 1 Pochhammer 3 2 m 1 -1 z 1 m 3 2 HypergeometricPFQ 1 m -1 ν 3 2 m ν 3 2 m 2 m 5 2 z 1 2 -1 m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "m"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["1", "+", "m"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "m"]], ",", RowBox[List["2", "+", "m"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["3", "2"], "+", "m"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List[FractionBox["5", "2"], "+", "m"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02