Generalized Laguerre polynomials: Specific values (formula 05.08.03.0019)

LaguerreL

$Mathematica Notation$

Polynomials

LaguerreL[n,lambda,z]

Specific values

Specialized values

For fixed lambda, z

http://functions.wolfram.com/05.08.03.0019.01

Input Form

LaguerreL[10, \[Lambda], z] == (1/3628800) (z^10 - 10 z^9 (10 + \[Lambda]) + 45 z^8 (90 + 19 \[Lambda] + \[Lambda]^2) - 120 z^7 (720 + 242 \[Lambda] + 27 \[Lambda]^2 + \[Lambda]^3) + 210 z^6 (5040 + 2414 \[Lambda] + 431 \[Lambda]^2 + 34 \[Lambda]^3 + \[Lambda]^4) - 252 z^5 (30240 + 19524 \[Lambda] + 5000 \[Lambda]^2 + 635 \[Lambda]^3 + 40 \[Lambda]^4 + \[Lambda]^5) + 210 z^4 (151200 + 127860 \[Lambda] + 44524 \[Lambda]^2 + 8175 \[Lambda]^3 + 835 \[Lambda]^4 + 45 \[Lambda]^5 + \[Lambda]^6) - 120 z^3 (604800 + 662640 \[Lambda] + 305956 \[Lambda]^2 + 77224 \[Lambda]^3 + 11515 \[Lambda]^4 + 1015 \[Lambda]^5 + 49 \[Lambda]^6 + \[Lambda]^7) + 45 z^2 (1814400 + 2592720 \[Lambda] + 1580508 \[Lambda]^2 + 537628 \[Lambda]^3 + 111769 \[Lambda]^4 + 14560 \[Lambda]^5 + 1162 \[Lambda]^6 + 52 \[Lambda]^7 + \[Lambda]^8) - 10 z (3628800 + 6999840 \[Lambda] + 5753736 \[Lambda]^2 + 2655764 \[Lambda]^3 + 761166 \[Lambda]^4 + 140889 \[Lambda]^5 + 16884 \[Lambda]^6 + 1266 \[Lambda]^7 + 54 \[Lambda]^8 + \[Lambda]^9) + 3628800 + 10628640 \[Lambda] + 12753576 \[Lambda]^2 + 8409500 \[Lambda]^3 + 3416930 \[Lambda]^4 + 902055 \[Lambda]^5 + 157773 \[Lambda]^6 + 18150 \[Lambda]^7 + 1320 \[Lambda]^8 + 55 \[Lambda]^9 + \[Lambda]^10)

Standard Form

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MathML Form

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<mo> + </mo> <mn> 151200 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> <mo> + </mo> <mrow> <mn> 49 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1015 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11515 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 77224 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 305956 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 662640 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 604800 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> 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<mi> λ </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1266 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16884 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 140889 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 761166 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2655764 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5753736 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6999840 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 3628800 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> λ </mi> <mn> 10 </mn> </msup> <mo> + </mo> <mrow> <mn> 55 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1320 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18150 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 157773 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 902055 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3416930 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8409500 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12753576 </mn> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10628640 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 3628800 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LaguerreL </ci> <cn type='integer'> 10 </cn> <ci> λ </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3628800 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 10 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 19 </cn> <ci> λ </ci> </apply> <cn type='integer'> 90 </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'> 120 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 242 </cn> <ci> λ </ci> </apply> <cn type='integer'> 720 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 210 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 34 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 431 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2414 </cn> <ci> λ </ci> </apply> <cn type='integer'> 5040 </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'> 252 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 635 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5000 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 19524 </cn> <ci> λ </ci> </apply> <cn type='integer'> 30240 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 210 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 835 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8175 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 44524 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 127860 </cn> <ci> λ </ci> </apply> <cn type='integer'> 151200 </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'> 120 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> <apply> <times /> <cn type='integer'> 49 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1015 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11515 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 77224 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 305956 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 662640 </cn> <ci> λ </ci> </apply> <cn type='integer'> 604800 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> 52 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1162 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14560 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 111769 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times 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type='integer'> 18150 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 157773 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 902055 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3416930 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8409500 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12753576 </cn> <apply> <power /> <ci> λ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10628640 </cn> <ci> λ </ci> </apply> <cn type='integer'> 3628800 </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]