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15.2 Legendre Polynomials 521

1

0.5

0
–1 –0.5 0 0.5 1
x

–0.5

–1

FIGURE 15.3 The first six Legendre
polynomials.

Graphs of these first six are shown in Figure 15.3.
In summary, the numbers λ = n(n + 1) for n = 0,1,2,··· are eigenvalues of Legendre’s
equation, for each nonnegative integer n. For each such n, ϕ n (x) = P n (x) is an eigenfunction.
This eigenfunction is an odd polynomial if n is odd and an even polynomial if n is even.
From Theorem 15.1, these Legendre polynomials (eigenfunctions) are orthogonal on [−1,1]
with respect to the weight function p(x) = 1. This means that

b
P n (x)P m (x)dx = 0
a
if n and m are distinct nonnegative integers. The weight function p(x) = 1 can be read directly
from the coefficient of y in Legendre’s differential equation.
There is an extensive literature on Legendre polynomials. We will develop some frequently
used facts about them.

15.2.1 A Generating Function for Legendre Polynomials

Define
1
L(x,t) = √ .
1 − 2xt + t 2
For a given x, we can think of L(x,t) as a function of t which can be expanded in a power
n
series about t = 0. We claim that, when this is done, the coefficient of t is P n (x). For this
reason we call L(x,t) a generating function for the Legendre polynomials.

THEOREM 15.3 A Generating Function

∞
n
L(x,t) = P n (x)t .
n=0

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