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
Read more about this topic: Legendre Polynomials
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