Rodrigues' Formula: The Legendre Polynomials can be expressed by Rodrigues' formula
Generating Function: The generating function of a Legendre Polynomial is
Orthogonality: Legendre Polynomials , , form a complete orthogonal set on the interval . It can be shown that
By using this orthogonality, a piecewise continuous function in can be expressed in terms of Legendre Polynomials:
This orthogonal series expansion is also known as a Fourier-Legendre Series expansion or a Generalized Fourier Series expansion.
Even/Odd Functions: Whether a Legendre Polynomial is an even or odd function depends on its degree .
Based on ,
is an even function when is even.
is an odd function when is odd.
In addition, from,
is an even function when is odd.
is an odd function when is even.
Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighboring Legendre Polynomials at the same point.