Legendre Polynomials - Additional Properties of Legendre Polynomials

Additional Properties of Legendre Polynomials

Legendre polynomials are symmetric or antisymmetric, that is

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that

The derivative at the end point is given by

As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula

and

Useful for the integration of Legendre polynomials is

From the above one can see also that

or equivalently

where is the norm over the interval −1 ≤ x ≤ 1

From Bonnet’s recursion formula one obtains by induction the explicit representation

The Askey–Gasper inequality for Legendre polynomials reads