Laguerre polynomials

Up to constant factors, these are the Laguerre polynomials.

The plain Laguerre polynomials are a subclass of these.

The Laguerre polynomials are orthogonal with respect to the exponential distribution.

These are the first few Laguerre polynomials:

This article relies heavily on Bessel functions and Laguerre polynomials.

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.)

Chapter 10 deals with Laguerre polynomials.

After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials.

The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials:

For further details, including the expressions for the first few polynomials, see Laguerre polynomials.

The sequence of Laguerre polynomials is a Sheffer sequence.

The addition formula for Laguerre polynomials:

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials.

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

Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

He also investigated orthogonal polynomials (see Laguerre polynomials).

It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r).

The generalized Laguerre polynomials are related to the Hermite polynomials:

(the classical Laguerre polynomials correspond to .)

The associated Laguerre polynomials are orthogonal over 0, with respect to the measure with weighting function x e:

Considering the problem in cylindrical coordinates, one can write higher-order modes using Laguerre polynomials instead of Hermite polynomials.

This example shows how the Hermite polynomials and Laguerre polynomials are interrelated through the Wigner-Weyl transform.

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

In mathematics, the Konhauser polynomials, introduced by , are biorthogonal polynomials for the distribution function of the Laguerre polynomials.