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http://functions.wolfram.com/05.08.23.0004.01
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Sum[Binomial[n + \[Lambda], n - k] LaguerreL[k, \[Lambda], z] w^k
(1 - w)^(n - k), {k, 0, n}] == LaguerreL[n, \[Lambda], z w]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "\[Lambda]"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", "\[Lambda]", ",", "z"]], "]"]], " ", SuperscriptBox["w", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["n", "-", "k"]]]]]]], "\[Equal]", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["z", " ", "w"]]]], "]"]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mi> λ </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "+", "\[Lambda]"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["n", "-", "k"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <msubsup> <mi> L </mi> <mi> k </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <ci> λ </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> k </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> λ </ci> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "n_"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n_", "+", "\[Lambda]_"]], ",", RowBox[List["n_", "-", "k_"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["k_", ",", "\[Lambda]_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "k_"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w_"]], ")"]], RowBox[List["n_", "-", "k_"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["z", " ", "w"]]]], "]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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