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Special Families of Polynomials 7
1.4 Legendre polynomials
Legendre polynomials are Jacobi ultraspherical polynomials with α = β = 0. To simplify
the notation it is standard to set Pn := P (0,0)
n . We now review the basic properties.
According to theorem 1.3.1, we have the differential equation
(1.4.1) (1 − x 2 ) P ′′
n − 2xP ′ n + n(n + 1)Pn = 0 , n ∈ N.
Conditions (1.3.2), (1.3.3) give respectively Pn (1) = 1, Pn (−1) = (−1) n , n ∈ N. The
recursion formula is
(1.4.2) Pn(x) =
2n − 1
n
xPn−1(x) −
n − 1
n Pn−2(x), x ∈ Ī, n ≥ 2.
Figure 1.4.1 - Legendre polynomials Figure 1.4.2 - The eleventh Legendre
for 1 ≤ n ≤ 6. polynomial.
It is easy to check that Pn is an even (odd) function if and only if n is even (odd).
Moreover, from (1.4.1) one gets
(1.4.3) P ′ n(±1) = ±
n(n + 1)
2
(±1) n , n ∈ N.