# Laguerre polynomials: laguerre

### 5.30.3 Laguerre polynomials: laguerre

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−xdx

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.

- Input:
  laguerre(2)
  Output:

  1
  2

  a2−a x+

  3
  2

  a+

  1
  2

  x2−2 x+1

- Input:
  laguerre(2,y)
  Output:

  1
  2

  a2−a y+

  3
  2

  a+

  1
  2

  y2−2 y+1

- Input:
  laguerre(2,y,b)
  Output:

  1
  2

  b2−b y+

  3
  2

  b+

  1
  2

  y2−2 y+1