### 5.27.1 Legendre polynomials: legendre

legendre takes as argument an integer n and optionally a variable name (by default x).
legendre returns the Legendre polynomial of degree n : it is a polynomial L(n,x), solution of the differential equation:

(x2−1) y″−2 x y′−n(n+1) y=0

The Legendre polynomials verify the following 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

Input :

legendre(4)

Output :

(35*x`^`4+-30*x`^`2+3)/8

Input :

legendre(4,y)

Output :

(35*y`^`4+-30*y`^`2+3)/8