### 11.4.1 Legendre polynomials

The Legendre polynomial L(n,x) of degree n is a polynomial solution of the differential equation

(x2−1)y′′−2x y′−n(n+1)y=0.

The Legendre polynomials satisfy the recurrence relation:

L(0,x) | =1 |  |  |  |  |  |  |  |  |
L(1,x) | =x |  |  |  |  |  |  |  |  |

L(n,x)

=

2n−1
n

x L(n−1,x)−

n−1
n

L(n−2,x)

These polynomials are orthogonal for the scalar product:

⟨f,g⟩=

∫

1
−1

f(x)g(x)d x.

The legendre command finds the Legendre polynomials.

- legendre takes one mandatory argument and one optional argument:
  - n, an integer.
  - Optionally, x, a variable name (by default x).
- legendre(n ⟨,x⟩) returns the Legendre polynomial of degree n.

#### Examples

legendre(4)

35
8

x4−

15
4

x2+

3
8

legendre(4,y)

35
8

y4−

15
4

y2+

3
8