# Hermite polynomial: hermite

### 5.30.2 Hermite polynomial: hermite

The Hermite polynomials H(n,x) satisfy the recurrence relation:

H(0,x) | =1 |  |  |  |  |  |  |  |  |
H(1,x) | =2x |  |  |  |  |  |  |  |  |
H(n,x)=2xH(n−1,x)−2(n−1)H(n−2,x) |  |  |  |  |  |  |  |  |  |

These polynomials are orthogonal for the scalar product:

<f,g>=

∫

+∞
−∞

f(x)g(x)e−x2dx

The hermite command finds the Hermite polynomials.

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

#### Examples.

- Input:
  hermite(6)
  Output:

  64 x6−480 x4+720 x2−120

- Input:
  hermite(6,y)
  Output:

  64 y6−480 y4+720 y2−120