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 ChebyshevU

 http://functions.wolfram.com/07.05.06.0057.01

 Input Form

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

 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[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "+", "2"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ",", RowBox[List["m", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", "1"]], ",", RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", "2"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", FractionBox["3", "2"]]], ",", RowBox[List["m", "+", "2"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "+", "3"]]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], ",", RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", FractionBox["5", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", "2"]], ",", RowBox[List["m", "+", FractionBox["5", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]]]]]]]], ")"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 U ν ( z ) F ( z , ν ) /; ( ( F m ( z , ν ) cos ( π ν 2 ) k = 0 m ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( ν 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! z 2 k + z ( ν + 1 ) sin ( π ν 2 ) k = 0 m ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! z 2 k U ν ( z ) - z 2 m + 2 cos ( π ν 2 ) ( ν 2 + 1 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( - ν 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( m + 1 ) ! ( 1 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] 3 F 2 ( 1 , m - ν 2 + 1 , m + ν 2 + 2 ; m + 3 2 , m + 2 ; z 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", "-", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", "2"]], 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[SuperscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z 2 m + 3 ( ν + 1 ) sin ( π ν 2 ) ( 1 - ν 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( ν + 3 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( m + 1 ) ! ( 3 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] 3 F 2 ( 1 , m - ν 2 + 3 2 , m + ν 2 + 5 2 ; m + 2 , m + 5 2 ; z 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", "-", FractionBox["\[Nu]", "2"], "+", FractionBox["3", "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", "+", FractionBox["5", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) m ) Condition ChebyshevU ν z Subscript F z ν Subscript F m z ν ν 2 -1 k 0 m Pochhammer -1 ν 2 -1 k Pochhammer ν 2 -1 1 k Pochhammer 1 2 k k -1 z 2 k z ν 1 ν 2 -1 k 0 m Pochhammer 1 -1 ν 2 -1 k Pochhammer ν 3 2 -1 k Pochhammer 3 2 k k -1 z 2 k ChebyshevU ν z -1 z 2 m 2 ν 2 -1 Pochhammer ν 2 -1 1 m 1 Pochhammer -1 ν 2 -1 m 1 m 1 Pochhammer 1 2 m 1 -1 HypergeometricPFQ 1 m -1 ν 2 -1 1 m ν 2 -1 2 m 3 2 m 2 z 2 -1 z 2 m 3 ν 1 ν 2 -1 Pochhammer 1 -1 ν 2 -1 m 1 Pochhammer ν 3 2 -1 m 1 m 1 Pochhammer 3 2 m 1 -1 HypergeometricPFQ 1 m -1 ν 2 -1 3 2 m ν 2 -1 5 2 m 2 m 5 2 z 2 m [/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[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", 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[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "+", "2"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ",", RowBox[List["m", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", "1"]], ",", RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", "2"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", FractionBox["3", "2"]]], ",", RowBox[List["m", "+", "2"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "+", "3"]]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["m", "-", FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], ",", RowBox[List["m", "+", FractionBox["\[Nu]", "2"], "+", FractionBox["5", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", "2"]], ",", RowBox[List["m", "+", FractionBox["5", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["m", "+", "1"]]]], "]"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02

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