Laguerre polynomials

### 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 dx.

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
| x2−2 x+1

laguerre(2,y)

1
2
| a2−a y+ |
3
2
| a+ |
1
2
| y2−2 y+1

laguerre(2,y,b)

1
2
| b2−b y+ |
3
2
| b+ |
1
2
| y2−2 y+1