The Legendre polynomials *P*_{n}(*x*) of degree *n* are special orthogonal polynomial functions defined on the domain [-1, 1].

Orthogonality:

_{-1}∫^{1} *W*(*x*) *P*_{i}(*x*) *P*_{j}(*x*) *dx* = δ_{ij} |

They obey certain recurrence relations:

*n* *P*_{n}(*x*) = (2*n* – 1) *x* *P*_{n-1}(*x*) – (*n* – 1) *P*_{n-2}(*x*) |

We may *change* the domain of these polynomials to [*xmin*, *xmax*] by using the following transformation:

*x*′ = (2*x* – (*xmax* + *xmin*)) / (*xmax* - *xmin*). |

We subsequently use *P*_{k}(*x*′) instead of *P*_{k}(*x*).

### Links to this page

© djmw, June 20, 1999