# M3: Laguerre Polynomials

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 $${\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,$$