# M3: Laguerre Polynomials

- Page ID
- 13500

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