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http://functions.wolfram.com/07.03.23.0007.01
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Sum[(LaguerreL[n, \[Lambda], z]/Pochhammer[\[Lambda] + 1, n]) w^n,
{n, 0, Infinity}] == E^w Hypergeometric0F1[\[Lambda] + 1, (-z) w] /;
Abs[w] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "n"]], "]"]]], " ", SuperscriptBox["w", "n"]]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List[RowBox[List["-", "z"]], " ", "w"]]]], "]"]]]]]], " ", "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]
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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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mi> n </mi> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> w </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", "1"]], Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", "z"]], " ", "w"]], Hypergeometric0F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> w </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <apply> <ci> Hypergeometric0F1 </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "n_"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]_", "+", "1"]], ",", "n_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List[RowBox[List["-", "z"]], " ", "w"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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