Hermite polynomial

### 11.4.2 Hermite polynomial

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−x2 dx.

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

hermite(6)

64 x6−480 x4+720 x2−120 |  |  |  |  |  |  |  |  |  |

hermite(6,y)

64 y6−480 y4+720 y2−120 |  |  |  |  |  |  |  |  |  |