Laguerre Polynomial
 Laguerre Polynomial
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 Definition 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 .
 Important Properties 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 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. •  •  •
Special Results

 where is 0 order Bessel function of the first kind.