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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,lambda,z] > Specific values > Specialized values > For fixed lambda, z





http://functions.wolfram.com/05.08.03.0015.01









  


  








Input Form





LaguerreL[6, \[Lambda], z] == (1/720) (z^6 - 6 z^5 (6 + \[Lambda]) + 15 z^4 (30 + 11 \[Lambda] + \[Lambda]^2) - 20 z^3 (120 + 74 \[Lambda] + 15 \[Lambda]^2 + \[Lambda]^3) + 15 z^2 (360 + 342 \[Lambda] + 119 \[Lambda]^2 + 18 \[Lambda]^3 + \[Lambda]^4) - 6 z (720 + 1044 \[Lambda] + 580 \[Lambda]^2 + 155 \[Lambda]^3 + 20 \[Lambda]^4 + \[Lambda]^5) + 720 + 1764 \[Lambda] + 1624 \[Lambda]^2 + 735 \[Lambda]^3 + 175 \[Lambda]^4 + 21 \[Lambda]^5 + \[Lambda]^6)








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mi> L </mi> <mn> 6 </mn> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 720 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 11 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 30 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 74 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 120 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#955; </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 18 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 119 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 342 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 360 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#955; </mi> <mn> 5 </mn> </msup> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 155 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 580 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1044 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 720 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> &#955; </mi> <mn> 6 </mn> </msup> <mo> + </mo> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 175 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 735 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1624 </mn> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1764 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 720 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LaguerreL </ci> <cn type='integer'> 6 </cn> <ci> &#955; </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 720 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 11 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 30 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <plus /> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 74 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 120 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 119 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 342 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 360 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 155 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 580 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1044 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 720 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 175 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 735 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1624 </cn> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1764 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 720 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["6", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "720"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "6"], "-", RowBox[List["6", " ", SuperscriptBox["z", "5"], " ", RowBox[List["(", RowBox[List["6", "+", "\[Lambda]"]], ")"]]]], "+", RowBox[List["15", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["30", "+", RowBox[List["11", " ", "\[Lambda]"]], "+", SuperscriptBox["\[Lambda]", "2"]]], ")"]]]], "-", RowBox[List["20", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["120", "+", RowBox[List["74", " ", "\[Lambda]"]], "+", RowBox[List["15", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", SuperscriptBox["\[Lambda]", "3"]]], ")"]]]], "+", RowBox[List["15", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["360", "+", RowBox[List["342", " ", "\[Lambda]"]], "+", RowBox[List["119", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["18", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", SuperscriptBox["\[Lambda]", "4"]]], ")"]]]], "-", RowBox[List["6", " ", "z", " ", RowBox[List["(", RowBox[List["720", "+", RowBox[List["1044", " ", "\[Lambda]"]], "+", RowBox[List["580", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["155", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["20", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", SuperscriptBox["\[Lambda]", "5"]]], ")"]]]], "+", "720", "+", RowBox[List["1764", " ", "\[Lambda]"]], "+", RowBox[List["1624", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["735", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["175", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", RowBox[List["21", " ", SuperscriptBox["\[Lambda]", "5"]]], "+", SuperscriptBox["\[Lambda]", "6"]]], ")"]]]]]]]]


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





2001-10-29