Generalized Laguerre polynomials: Specific values (formula 05.08.03.0018)

LaguerreL

$Mathematica Notation$

Polynomials

LaguerreL[n,lambda,z]

Specific values

Specialized values

For fixed lambda, z

http://functions.wolfram.com/05.08.03.0018.01

Input Form

LaguerreL[9, \[Lambda], z] == (1/362880) (-z^9 + 9 z^8 (9 + \[Lambda]) - 36 z^7 (72 + 17 \[Lambda] + \[Lambda]^2) + 84 z^6 (504 + 191 \[Lambda] + 24 \[Lambda]^2 + \[Lambda]^3) - 126 z^5 (3024 + 1650 \[Lambda] + 335 \[Lambda]^2 + 30 \[Lambda]^3 + \[Lambda]^4) + 126 z^4 (15120 + 11274 \[Lambda] + 3325 \[Lambda]^2 + 485 \[Lambda]^3 + 35 \[Lambda]^4 + \[Lambda]^5) - 84 z^3 (60480 + 60216 \[Lambda] + 24574 \[Lambda]^2 + 5265 \[Lambda]^3 + 625 \[Lambda]^4 + 39 \[Lambda]^5 + \[Lambda]^6) + 36 z^2 (181440 + 241128 \[Lambda] + 133938 \[Lambda]^2 + 40369 \[Lambda]^3 + 7140 \[Lambda]^4 + 742 \[Lambda]^5 + 42 \[Lambda]^6 + \[Lambda]^7) - 9 z (362880 + 663696 \[Lambda] + 509004 \[Lambda]^2 + 214676 \[Lambda]^3 + 54649 \[Lambda]^4 + 8624 \[Lambda]^5 + 826 \[Lambda]^6 + 44 \[Lambda]^7 + \[Lambda]^8) + 362880 + 1026576 \[Lambda] + 1172700 \[Lambda]^2 + 723680 \[Lambda]^3 + 269325 \[Lambda]^4 + 63273 \[Lambda]^5 + 9450 \[Lambda]^6 + 870 \[Lambda]^7 + 45 \[Lambda]^8 + \[Lambda]^9)

Standard Form

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MathML Form

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</mo> <mrow> <msup> <mi> λ </mi> <mn> 8 </mn> </msup> <mo> + </mo> <mrow> <mn> 44 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 826 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8624 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 54649 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 214676 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 509004 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 663696 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 362880 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> λ </mi> <mn> 9 </mn> </msup> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 870 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9450 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 63273 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 269325 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 723680 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1172700 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1026576 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 362880 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LaguerreL </ci> <cn type='integer'> 9 </cn> <ci> λ </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 362880 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 17 </cn> <ci> λ </ci> </apply> <cn type='integer'> 72 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 191 </cn> <ci> λ </ci> </apply> <cn type='integer'> 504 </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'> 126 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 335 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1650 </cn> <ci> λ </ci> </apply> <cn type='integer'> 3024 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 126 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 485 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3325 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11274 </cn> <ci> λ </ci> </apply> <cn type='integer'> 15120 </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'> 84 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> 39 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 625 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5265 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24574 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 60216 </cn> <ci> λ </ci> </apply> <cn type='integer'> 60480 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> <apply> <times /> <cn type='integer'> 42 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 742 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7140 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 40369 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 133938 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 241128 </cn> <ci> λ </ci> </apply> <cn type='integer'> 181440 </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'> 9 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> 44 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 826 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8624 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 54649 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 214676 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 509004 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 663696 </cn> <ci> λ </ci> </apply> <cn type='integer'> 362880 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> λ </ci> <cn type='integer'> 9 </cn> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 870 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9450 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 63273 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 269325 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 723680 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1172700 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1026576 </cn> <ci> λ </ci> </apply> <cn type='integer'> 362880 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

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