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http://functions.wolfram.com/07.03.20.0007.01
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D[LaguerreL[\[Nu], \[Lambda], z], {\[Lambda], 2}] ==
(-(2/Pi)) Sin[\[Nu] Pi] Gamma[\[Lambda] + \[Nu] + 1]^3
Sum[(z^k/k!) HypergeometricPFQRegularized[{\[Lambda] + \[Nu] + 1,
\[Lambda] + \[Nu] + 1, \[Lambda] + \[Nu] + 1, 1 + \[Nu] - k},
{\[Lambda] + \[Nu] + 2, \[Lambda] + \[Nu] + 2, \[Lambda] + \[Nu] + 2},
1], {k, 0, Infinity}] /; !(Element[\[Nu], Integers] && \[Nu] >= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Lambda]", ",", "2"]], "}"]]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[" ", "2"]], "\[Pi]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], "]"]], "3"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["1", "+", "\[Nu]", "-", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]]]], "}"]], ",", "1"]], "]"]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], "]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <mrow> <msubsup> <mi> L </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <msup> <mi> z </mi> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 4 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "k", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "\[Lambda]", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mi> ν </mi> <mo> ∉ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> λ </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> LaguerreL </ci> <ci> ν </ci> <ci> λ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sin /> <apply> <times /> <ci> ν </ci> <pi /> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> ν </ci> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </list> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> ν </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Lambda]_", ",", "2"]], "}"]]]]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], "]"]], "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], ",", RowBox[List["1", "+", "\[Nu]", "-", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]], ",", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "2"]]]], "}"]], ",", "1"]], "]"]]]], RowBox[List["k", "!"]]]]]]], "\[Pi]"]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
