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2.26.1  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)=
2n−1
n
x L(n−1,x)−
n−1
n
L(n−2,x)

These polynomials are orthogonal for the scalar product :

<f,g>=
+1


−1
f(x)g(xdx 

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

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