Rodrigues' Formula: The Legendre Polynomials can be expressed by Rodrigues' formula
where
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:
where:
This orthogonal series expansion is also known as a FourierLegendre 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.
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