Hermite polynomial : hermite

hermite takes as argument an integer n and optionnally a variable name (by default x).
hermite returns the Hermite polynomial of degree n.
If H(n, x) denotes the Hermite polynomial of degree n, the following recurrence relation holds:

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 > = $$\int_{{-\infty}}^{{+\infty}}$$f (x)g(x)e^-x2dx

Input :

hermite(6)

Output :

64*x^6+-480*x^4+720*x^2-120

Input :

hermite(6,y)

Output :

64*y^6+-480*y^4+720*y^2-120