Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation:
which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
More generally, the name Laguerre polynomials is used for solutions of
Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventorNikolay Yakovlevich Sonin).
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,
reducing to the closed form of a following section.
They are orthogonal polynomials with respect to an inner product
The sequence of Laguerre polynomials n! Ln is a Sheffer sequence,
The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials.
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator.
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