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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0068.01

 Input Form

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

 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[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]], " ", "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Nu]", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "+", "\[Nu]"]]], " ", "\[Nu]", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "-", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "-", "\[Nu]"]]], " ", "\[Nu]", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 T ν ( z ) F ( z , ν ) /; ( F m ( z , ν ) 2 - ν - 1 z - ν k = 0 m ( ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Nu]", "2"], ")"]], "k"], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! z - 2 k + 2 ν - 1 z ν k = 0 m ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! z - 2 k T ν ( z ) - 2 - 2 m - ν - 3 z - 2 ( m + 1 ) - ν ν Γ ( 2 m + ν + 2 ) ( m + 1 ) ! Γ ( m + ν + 2 ) 3 F 2 ( 1 , m + ν 2 + 1 , m + ν 2 + 3 2 ; m + 2 , m + ν + 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", 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", "+", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 - 2 m + ν - 3 z ν - 2 ( m + 1 ) ν Γ ( 2 m - ν + 2 ) ( m + 1 ) ! Γ ( m - ν + 2 ) 3 F 2 ( 1 , m - ν 2 + 1 , m - ν 2 + 3 2 ; m + 2 , m - ν + 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", 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", "-", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] m ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition ChebyshevT ν z Subscript F z ν Subscript F m z ν 2 -1 ν -1 z -1 ν k 0 m Pochhammer ν 2 -1 k Pochhammer ν 1 2 -1 k Pochhammer ν 1 k k -1 z -2 k 2 ν -1 z ν k 0 m Pochhammer -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k Pochhammer 1 -1 ν k k -1 z -2 k ChebyshevT ν z -1 2 -2 m -1 ν -3 z -2 m 1 -1 ν ν Gamma 2 m ν 2 m 1 Gamma m ν 2 -1 HypergeometricPFQ 1 m ν 2 -1 1 m ν 2 -1 3 2 m 2 m ν 2 1 z 2 -1 2 -2 m ν -3 z ν -1 2 m 1 ν Gamma 2 m -1 ν 2 m 1 Gamma m -1 ν 2 -1 HypergeometricPFQ 1 m -1 ν 2 -1 1 m -1 ν 2 -1 3 2 m 2 m -1 ν 2 1 z 2 -1 m ν [/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[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Nu]", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "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["-", "3"]], "-", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "+", "\[Nu]"]]], " ", "\[Nu]", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "-", "\[Nu]"]]], " ", "\[Nu]", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Nu]"]], "]"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2007-05-02