Legendre polynomials

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)

L(1,x)

L(n,x)

2n−1

x L(n−1,x)−

n−1

L(n−2,x)

These polynomials are orthogonal for the scalar product:

⟨ f,g⟩=

∫

−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.

legendre(4)

x⁴−

x²+

legendre(4,y)

y⁴−

y²+