Rodrigues' Formula: The Laguerre Polynomials can be expressed by Rodrigues' formula:
where
Generating Function: The generating function of a Laguerre Polynomial is:
Orthogonality: Laguerre Polynomials , , form a complete orthogonal set on the interval with respect to the weighting function . It can be shown that:
By using this orthogonality, a piecewise continuous function can be expressed in terms of Laguerre Polynomials:
where:
This orthogonal series expansion is also known as a FourierLaguerre Series expansion or a Generalized Fourier Series expansion.
Recurrence Relation: A Laguerre Polynomial at one point can be expressed in terms of neighboring Laguerre Polynomials at the same point.
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