sage: integrate(Pj*Pk, x, a, b) # abs tol 1e-12
0
+ The first few polynomials shifted to [0,1] are known to be::
+
+ sage: p0 = 1
+ sage: p1 = 2*x - 1
+ sage: p2 = 6*x^2 - 6*x + 1
+ sage: p3 = 20*x^3 - 30*x^2 + 12*x - 1
+ sage: bool(legendre_p(0, x, 0, 1) == p0)
+ True
+ sage: bool(legendre_p(1, x, 0, 1) == p1)
+ True
+ sage: bool(legendre_p(2, x, 0, 1) == p2)
+ True
+ sage: bool(legendre_p(3, x, 0, 1) == p3)
+ True
+
"""
if not n in ZZ:
raise TypeError('n must be a natural number')