# Legendre polynomials: legendre

### 5.30.1 Legendre polynomials: legendre

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

(x2−1) y″−2 x 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) dx

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.

- Input:
  legendre(4)
  Output:

  35
  8

  x4−

  15
  4

  x2+

  3
  8

- Input:
  legendre(4,y)
  Output:

  35
  8

  y4−

  15
  4

  y2+

  3
  8