Generalized Laguerre function: Specific values (formula 07.03.03.0017)

LaguerreL

$Mathematica Notation$

Hypergeometric Functions

LaguerreL[nu,lambda,z]

Specific values

Specialized values

For fixed lambda, z

http://functions.wolfram.com/07.03.03.0017.01

Input Form

LaguerreL[8, \[Lambda], z] == (1/40320) (z^8 - 8 z^7 (8 + \[Lambda]) + 28 z^6 (56 + 15 \[Lambda] + \[Lambda]^2) - 56 z^5 (336 + 146 \[Lambda] + 21 \[Lambda]^2 + \[Lambda]^3) + 70 z^4 (1680 + 1066 \[Lambda] + 251 \[Lambda]^2 + 26 \[Lambda]^3 + \[Lambda]^4) - 56 z^3 (6720 + 5944 \[Lambda] + 2070 \[Lambda]^2 + 355 \[Lambda]^3 + 30 \[Lambda]^4 + \[Lambda]^5) + 28 z^2 (20160 + 24552 \[Lambda] + 12154 \[Lambda]^2 + 3135 \[Lambda]^3 + 445 \[Lambda]^4 + 33 \[Lambda]^5 + \[Lambda]^6) - 8 z (40320 + 69264 \[Lambda] + 48860 \[Lambda]^2 + 18424 \[Lambda]^3 + 4025 \[Lambda]^4 + 511 \[Lambda]^5 + 35 \[Lambda]^6 + \[Lambda]^7) + 40320 + 109584 \[Lambda] + 118124 \[Lambda]^2 + 67284 \[Lambda]^3 + 22449 \[Lambda]^4 + 4536 \[Lambda]^5 + 546 \[Lambda]^6 + 36 \[Lambda]^7 + \[Lambda]^8)

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> 8 </mn> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 40320 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 56 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 56 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 146 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 336 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 70 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 26 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 251 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1066 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1680 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 56 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> <mo> + </mo> <mrow> <mn> 30 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 355 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2070 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5944 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 6720 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> <mo> + </mo> <mrow> <mn> 33 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 445 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3135 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12154 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 24552 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 20160 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> <mo> + </mo> <mrow> <mn> 35 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 511 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4025 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18424 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48860 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 69264 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 40320 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> λ </mi> <mn> 8 </mn> </msup> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 546 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4536 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 22449 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 67284 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 118124 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 109584 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 40320 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LaguerreL </ci> <cn type='integer'> 8 </cn> <ci> λ </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 40320 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <ci> λ </ci> </apply> <cn type='integer'> 56 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 146 </cn> <ci> λ </ci> </apply> <cn type='integer'> 336 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 70 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 26 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 251 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1066 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1680 </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'> 56 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 355 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2070 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5944 </cn> <ci> λ </ci> </apply> <cn type='integer'> 6720 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> 33 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 445 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3135 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12154 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24552 </cn> <ci> λ </ci> </apply> <cn type='integer'> 20160 </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'> 8 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 511 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4025 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18424 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 48860 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 69264 </cn> <ci> λ </ci> </apply> <cn type='integer'> 40320 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> λ </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 546 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4536 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 22449 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 67284 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 118124 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 109584 </cn> <ci> λ </ci> </apply> <cn type='integer'> 40320 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["8", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "8"], "-", RowBox[List["8", " ", SuperscriptBox["z", "7"], " ", RowBox[List["(", RowBox[List["8", "+", "\[Lambda]"]], ")"]]]], "+", RowBox[List["28", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List["56", "+", RowBox[List["15", " ", "\[Lambda]"]], "+", SuperscriptBox["\[Lambda]", "2"]]], ")"]]]], "-", RowBox[List["56", " ", SuperscriptBox["z", "5"], " ", RowBox[List["(", RowBox[List["336", "+", RowBox[List["146", " ", "\[Lambda]"]], "+", RowBox[List["21", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", SuperscriptBox["\[Lambda]", "3"]]], ")"]]]], "+", RowBox[List["70", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["1680", "+", RowBox[List["1066", " ", "\[Lambda]"]], "+", RowBox[List["251", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["26", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", SuperscriptBox["\[Lambda]", "4"]]], ")"]]]], "-", RowBox[List["56", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["6720", "+", RowBox[List["5944", " ", "\[Lambda]"]], "+", RowBox[List["2070", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["355", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["30", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", SuperscriptBox["\[Lambda]", "5"]]], ")"]]]], "+", RowBox[List["28", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["20160", "+", RowBox[List["24552", " ", "\[Lambda]"]], "+", RowBox[List["12154", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["3135", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["445", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", RowBox[List["33", " ", SuperscriptBox["\[Lambda]", "5"]]], "+", SuperscriptBox["\[Lambda]", "6"]]], ")"]]]], "-", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List["40320", "+", RowBox[List["69264", " ", "\[Lambda]"]], "+", RowBox[List["48860", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["18424", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["4025", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", RowBox[List["511", " ", SuperscriptBox["\[Lambda]", "5"]]], "+", RowBox[List["35", " ", SuperscriptBox["\[Lambda]", "6"]]], "+", SuperscriptBox["\[Lambda]", "7"]]], ")"]]]], "+", "40320", "+", RowBox[List["109584", " ", "\[Lambda]"]], "+", RowBox[List["118124", " ", SuperscriptBox["\[Lambda]", "2"]]], "+", RowBox[List["67284", " ", SuperscriptBox["\[Lambda]", "3"]]], "+", RowBox[List["22449", " ", SuperscriptBox["\[Lambda]", "4"]]], "+", RowBox[List["4536", " ", SuperscriptBox["\[Lambda]", "5"]]], "+", RowBox[List["546", " ", SuperscriptBox["\[Lambda]", "6"]]], "+", RowBox[List["36", " ", SuperscriptBox["\[Lambda]", "7"]]], "+", SuperscriptBox["\[Lambda]", "8"]]], "40320"]]]]]

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

2001-10-29

LaguerreL[n,z]
LaguerreL[nu,z]
LaguerreL[n,mu,z]