### 11.4.3 Laguerre polynomials

The Laguerre polynomial of degree n and parameter a satisfy the following recurrence relation:

L(0,a,x) | =1 |  |  |  |  |  |  |  |  |
L(1,a,x) | =1+a−x |  |  |  |  |  |  |  |  |

L(n,a,x)

=

2n+a−1−x
n

L(n−1,a,x)−

n+a−1
n

L(n−2,a,x)

These polynomials are orthogonal for the scalar product

⟨f,g⟩=

∫

+∞
0

f(x)g(x)xae−x d x.

The laguerre command finds the Laguerre polynomials.

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

#### Examples

laguerre(2)

1
2

a2−a x+

3
2

a+

1
2

x2−2x+1

laguerre(2,y)

1
2

a2−a y+

3
2

a+

1
2

y2−2y+1

laguerre(2,y,b)

1
2

b2−b y+

3
2

b+

1
2

y2−2y+1