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 ChebyshevU

 http://functions.wolfram.com/07.05.06.0077.01

 Input Form

 ChebyshevU[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; (Subscript[F, m][z, \[Nu]] == 2^\[Nu] z^\[Nu] Sum[(Pochhammer[-(\[Nu]/2), k] Pochhammer[(1 - \[Nu])/2, k])/(Pochhammer[-\[Nu], k] k!)/z^(2 k), {k, 0, m}] - 2^(-2 - \[Nu]) z^(-2 - \[Nu]) Sum[(Pochhammer[(3 + \[Nu])/2, k] Pochhammer[(2 + \[Nu])/2, k])/ (Pochhammer[2 + \[Nu], k] k!)/z^(2 k), {k, 0, m}] == ChebyshevU[\[Nu], z] - ((2^(-2 - 2 m + \[Nu]) z^(-2 - 2 m + \[Nu]) Gamma[2 + 2 m - \[Nu]])/(Gamma[2 + m] Gamma[1 + m - \[Nu]])) HypergeometricPFQ[{1, 1 + m - \[Nu]/2, 3/2 + m - \[Nu]/2}, {2 + m, 1 + m - \[Nu]}, 1/z^2] + ((2^(-4 - 2 m - \[Nu]) z^(-4 - 2 m - \[Nu]) Gamma[4 + 2 m + \[Nu]])/ (Gamma[2 + m] Gamma[3 + m + \[Nu]])) HypergeometricPFQ[ {1, 2 + m + \[Nu]/2, 5/2 + m + \[Nu]/2}, {2 + m, 3 + m + \[Nu]}, 1/z^2] && Element[m, Integers] && m >= 0) && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevU", "[", 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", "\[Nu]"], " ", 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["-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]], " ", "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", "m"]], "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "+", "m"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "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["1", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", "m"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["4", "+", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "+", "m"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "m", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "m", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "2"], "+", "m", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["3", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], ")"]], "\[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

 U ν ( z ) F ( z , ν ) /; ( F m ( z , ν ) 2 ν 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] ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! z - 2 k - 2 - ν - 2 z - ν - 2 k = 0 m ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ")"]], "k"], Pochhammer] ( ν + 2 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], ")"]], "k"], Pochhammer] ( ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "2"]], ")"]], "k"], Pochhammer] k ! z - 2 k U ν ( z ) + 2 - 2 m - ν - 4 z - 2 m - ν - 4 Γ ( 2 m + ν + 4 ) Γ ( m + 2 ) Γ ( m + ν + 3 ) 3 F 2 ( 1 , m + ν 2 + 2 , m + ν 2 + 5 2 ; m + 2 , m + ν + 3 ; 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"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", FractionBox["5", "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]", "+", "3"]], 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 - 2 z - 2 m + ν - 2 Γ ( 2 m - ν + 2 ) Γ ( m + 2 ) Γ ( m - ν + 1 ) 3 F 2 ( 1 , m - ν 2 + 1 , m - ν 2 + 3 2 ; m + 2 , m - ν + 1 ; 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]", "+", "1"]], 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 ChebyshevU ν z Subscript F z ν Subscript F m z ν 2 ν z ν k 0 m Pochhammer -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k Pochhammer -1 ν k k -1 z -2 k -1 2 -1 ν -2 z -1 ν -2 k 0 m Pochhammer ν 3 2 -1 k Pochhammer ν 2 2 -1 k Pochhammer ν 2 k k -1 z -2 k ChebyshevU ν z 2 -2 m -1 ν -4 z -2 m -1 ν -4 Gamma 2 m ν 4 Gamma m 2 Gamma m ν 3 -1 HypergeometricPFQ 1 m ν 2 -1 2 m ν 2 -1 5 2 m 2 m ν 3 1 z 2 -1 -1 2 -2 m ν -2 z -2 m ν -2 Gamma 2 m -1 ν 2 Gamma m 2 Gamma m -1 ν 1 -1 HypergeometricPFQ 1 m -1 ν 2 -1 1 m -1 ν 2 -1 3 2 m 2 m -1 ν 1 1 z 2 -1 m ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevU", "[", 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", "\[Nu]"], " ", 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["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "m"]], "+", "\[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["1", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "+", "m"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "-", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["4", "+", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "m", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "2"], "+", "m", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["3", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "+", "m"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "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