Legendre polynomials

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

Orthogonality:

-11 W(x) Pi(x) Pj(x) dx = δij
W(x) = 1 (-1 < x < 1)

They obey certain recurrence relations:

n Pn(x) = (2n – 1) x Pn-1(x) – (n – 1) Pn-2(x)
P0(x) = 1
P1(x) = x

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

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

We subsequently use Pk(x′) instead of Pk(x).

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© djmw, June 20, 1999