Laguerre Polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886), are solutions of Laguerre's equation:

$x\,y'' + (1 - x)\,y' + n\,y = 0\,$

which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials (also named Sonin polynomials after Nikolay Yakovlevich Sonin in some older books) are solutions of

$x\,y'' + (\alpha+1 - x)\,y' + n\,y = 0\,$

The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form

These polynomials, usually denoted L₀, L₁, ..., are a polynomial sequence which may be defined by the Rodrigues formula

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).$

They are orthonormal to each other with respect to the inner product given by

The sequence of Laguerre polynomials is a Sheffer sequence.

The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables.

The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.

Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of n!, than the definition used here. (Furthermore, various physicist use somewhat different definitions of the so-called associated Laguerre polynomials, for instance in the definition is different than the one found below. A comparison of notations can be found in .)