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 LaguerreL

 http://functions.wolfram.com/07.03.06.0020.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]", ",", "\[Lambda]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]", ",", "\[Lambda]"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "k", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Lambda]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Lambda]"]], "]"]]]]], SuperscriptBox["z", RowBox[List["1", "+", "m"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Lambda]"]]]], "}"]], ",", "z"]], "]"]]]]]]]], ")"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 L ν λ ( z ) F ( z , ν , λ ) /; ( ( F m ( z , ν , λ ) Γ ( λ + ν + 1 ) Γ ( ν + 1 ) k = 0 m ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] z k Γ ( k + λ + 1 ) k ! L ν λ ( z ) + Γ ( m - ν + 1 ) Γ ( λ + ν + 1 ) sin ( π ν ) π ( m + 1 ) ! Γ ( m + λ + 2 ) z m + 1 2 F 2 ( 1 , m - ν + 1 ; m + 2 , m + λ + 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", 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]]]], 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", "+", "\[Lambda]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) m ) Condition LaguerreL ν λ z Subscript F z ν λ Subscript F m z ν λ Gamma λ ν 1 Gamma ν 1 -1 k 0 m Pochhammer -1 ν k z k Gamma k λ 1 k -1 LaguerreL ν λ z Gamma m -1 ν 1 Gamma λ ν 1 ν m 1 Gamma m λ 2 -1 z m 1 HypergeometricPFQ 1 m -1 ν 1 m 2 m λ 2 z m [/itex]

 Rule Form

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

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

 2007-05-02