M3: Laguerre Polynomials
- Page ID
- 13500
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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 | \({\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,\) |