M3: Laguerre Polynomials

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Sep 5, 2019
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- M2: Legendre Polynomials
- M4: Spherical Harmonics

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The Laguerre polynomials are solutions of Laguerre's differential equation:

$x\,y'' + (1 - x)\,y' + n\,y = 0\,$

These are the first few Laguerre polynomials:

$n$	$L_n(x)\,$
0	1
1	$-x+1\,$
2	${\frac{1}{2}} (x^2-4x+2) \,$
3	${\frac{1}{6}} (-x^3+9x^2-18x+6) \,$
4	${\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \,$
5	${\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \,$
6	${\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,$

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