Legendre polynomials: legendre

legendre takes as argument an integer n and optionnally a variable name (by default x).
legendre returns the Legendre polynomial of degree n : it is a polynomial L(n, x), solution of the differential equation:

(x² - 1).y'' - 2.x.y' - n(n + 1).y = 0

The Legendre polynomials verify the following recurrence relation:

L(0, x) = 1,    L(1, x) = x,    L(n, x) = $${\frac{{2n-1}}{{n}}}$$xL(n - 1, x) - $${\frac{{n-1}}{{n}}}$$L(n - 2, x)

These polynomials are orthogonal for the scalar product :

< f, g > = $$\int_{{-1}}^{{+1}}$$f (x)g(x) dx

Input :

legendre(4)

Output :

(35*x^4+-30*x^2+3)/8

Input :

legendre(4,y)

Output :

(35*y^4+-30*y^2+3)/8