In the Sturm-Liouville Boundary Value Problem, there is a special case called Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Laguerre's Differential Equation is defined as:
where is a real number. When is a non-negative integer, i.e., , the solutions of Laguerre's Differential Equation are often referred to as Laguerre Polynomials .
Rodrigues' Formula: The Laguerre Polynomials can be expressed by Rodrigues' formula:
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:
This orthogonal series expansion is also known as a Fourier-Laguerre 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.