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