which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
Sometimes the name Laguerre polynomials is used for solutions of
where n 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 n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form
reducing to the closed form of a following section.
The sequence of Laguerre polynomials n! Ln is a Sheffer sequence,
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 which 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.)
The first few polynomials
These are the first few Laguerre polynomials:
Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
In solution of some boundary value problems, the characteristic values can be useful:
The closed form is
The generating function for them likewise follows,
Polynomials of negative index can be expressed using the ones with positive index:
Generalized Laguerre polynomials
are called generalized Laguerre polynomials, or associated Laguerre polynomials.
One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
The simple Laguerre polynomials are the special case α = 0 of the generalized Laguerre polynomials:
The Rodrigues formula for them is
The generating function for them is
Explicit examples and properties of the generalized Laguerre polynomials
- Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as
- is a generalized binomial coefficient. When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression
- in terms of Kummer's function of the second kind.
- derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
- The first few generalized Laguerre polynomials are:
- If α is non-negative, then Ln(α) has n real, strictly positive roots (notice that is a Sturm chain), which are all in the interval
- The polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by
- and summarizing by
- where is the Bessel function.
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
Laguerre's polynomials satisfy the recurrence relations
They can be used to derive the four 3-point-rules
combined they give this additional, useful recurrence relations
Since is a monic polynomial of degree in , there is the partial fraction decomposition
The second equality follows by the following identity, valid for integer i and n and immediate from the expression of in terms of Charlier polynomials:
For the third equality apply the fourth and fifth identities of this section.
Derivatives of generalized Laguerre polynomials
Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
This points to a special case (α = 0) of the formula above: for integer α = k the generalized polynomial may be written
the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
Moreover, the following equation holds:
which generalizes with Cauchy's formula to
This is evident from the contour integral representation below.
The generalized Laguerre polynomials obey the differential equation
which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,
where for this equation only.
In Sturm–Liouville form the differential equation is
which shows that L(α)
n is an eigenvector for the eigenvalue n.
which follows from
If denotes the Gamma distribution then the orthogonality relation can be written as
The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)
Turán's inequalities can be derived here, which is
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
Let a function have the (formal) series expansion
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.
Relation to Hermite polynomials
The generalized Laguerre polynomials are related to the Hermite polynomials:
where the Hn(x) are the Hermite polynomials based on the weighting function exp(−x2), 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
where is the Pochhammer symbol (which in this case represents the rising factorial).
where the series on the left converges for and . Using the identity
(see generalized hypergeometric function), this can also be written as
- Transverse mode, an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile.
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