 Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that

The derivative at the end point is given by

The Askey–Gasper inequality for Legendre polynomials reads

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using

where the unit vectors and have spherical coordinates and , respectively.As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet’s recursion formula
and

or, with the alternative expression, which also holds at the endpoints

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 

Asymptotically for formula_88

and for arguments of magnitude greater than 1
where and are Bessel functions.All formula_91 zeros of formula_3 are real, distinct from each other, and lie in the interval formula_93.