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Wikipedia / Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer.

Sometimes the name Laguerre polynomials where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).

More generally, a Laguerre function is a solution when is not necessarily a non-negative integer.

The Laguerre polynomials are also used for Gauss–Laguerre quadrature

These polynomials, usually denoted ,,..., 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 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.

Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)

Recursive definition, closed form, and generating function

and then using the following recurrence relation for any

The closed form

The generating function

n
0
1
2
3
4
5
6
7
8
9
10
n

A table of the Laguerre polynomials

Generalized Laguerre polynomials

For arbitrary real α the polynomial solutions of the differential equation are called generalized Laguerre polynomials, or associated Laguerre polynomials.

and then using the following recurrence relation for any

The simple Laguerre polynomials are the special case

The Rodrigues formula

The generating function

Properties

Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as is a generalized binomial coefficient. When is an integer the function reduces to a polynomial of degree . It has the alternative expression in terms of Kummer's function of the second kind.
The closed form for these generalized Laguerre polynomials of degree is derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
Laguerre polynomials have a differential operator.
The first few generalized Laguerre polynomials are:

n
0
1
2
3
4
5
6
7
8
9
10

The coefficient of the leading term is ;
The constant term, which is the value at
The discriminant is

As a contour integral

Given the generating function specified above, the polynomials may be expressed in terms of a contour integral where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1

Recurrence relations

The addition formula for Laguerre polynomials:

is a monic polynomial, there is the partial fraction decomposition The second equality follows by the following identity, valid for integer i and in terms of Charlier polynomials For the third equality apply the fourth and fifth identities of this section.

Derivatives

Differentiating the power series representation of a generalized Laguerre polynomial

This points to a special case () of the formula above: for integer the shift by sometimes causing confusion with the usual parenthesis notation for a derivative.

which generalizes with Cauchy's formula

The derivative with respect to the second variable has the form, which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

for this equation only.

In Sturm–Liouville form the differential equation is

which shows that is an eigenvector for the eigenvalue .

Orthogonality

The generalized Laguerre polynomials are orthogonal over with respect to the measure with weighting function :

which follows from

denotes the gamma distribution then the orthogonality relation can be written as

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)

recursively

Turán's inequalities

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

Series expansions

The series converges in the associated Hilbert space if and only if

Further examples of expansions

Monomials while binomials

for the exponential function. The incomplete gamma function

Asymptotics

In terms of elementary functions

, fixed real numberThis is Perron's formula.. Fejér.

In terms of Bessel functions

The Mehler–Heine formula states:

is a Bessel function of the first kind.

In terms of Airy functions

be the Airy function be positive and fixed.

The Plancherel–Rotach asymptotics formulas:

:
:
:

See DLMF for higher-order terms.

Zeroes

Notation

-th positive zero of the Bessel function.

by default.

is a Sturm chain.

Inequalities

, we have these bounds:

,, so the first inequality is sharp.

Electrostatics

The zeroes satisfy the Stieltjes relations:, and produce a constant electric field are the equilibrium positions of the particles.

As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Laguerre polynomials.

The zeroes also satisfy

Limit distribution

be the cumulative distribution function for the roots, then we have the limit lawwhich can be interpreted as the limit distribution of the Wishart ensemble spectrum.

In quantum mechanics

In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.

Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.

Multiplication theorems

Erdélyi gives the following two multiplication theorems

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials where the are the Hermite polynomials based on the weighting function , the so-called "physicist's version."

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

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions is the Pochhammer symbol (which in this case represents the rising factorial).

Hardy–Hille formula

The generalized Laguerre polynomials satisfy the Hardy–Hille formula (see generalized hypergeometric functionThis formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by setting the Hermite polynomials as a special case of the associated Laguerre polynomials.

limit, we obtain The formula is named after G. H. Hardy and Einar Hille.

Physics convention

The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals. The convention used throughout this article expresses the generalized Laguerre polynomials as

is the confluent hypergeometric function. In the physics literature, the generalized Laguerre polynomials are instead defined as

The physics version is related to the standard version by

There is yet another, albeit less frequently used, convention in the physics literature

Umbral calculus convention

Generalized Laguerre polynomials are linked to umbral calculus by being Sheffer sequences. In umbral calculus convention, are the signless Lah numbers is a sequence of polynomials of binomial type, i.e.