Legendre polynomials: legendre

### 2.26.1 Legendre polynomials: `legendre`

`legendre` takes as argument an integer n and optionnally 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`