In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886), are the canonical 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. 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
They are orthogonal 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 [Modern Quantum mechanics by J.J. Sakurai] the definition is different than the one found below. A comparison of notations can be found in [Introductory quantum mechanics by R.L. Liboff].)
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These are the first few Laguerre polynomials:
n | |
0 | |
1 | |
2 | |
3 | |
4 | |
5 | |
6 |
We can also define the Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
The polynomial solution of differential equation[1]
is called generalized Laguerre polynomials, or associated Laguerre polynomials. The Rodrigues' formula for them are
The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:
where is the Bessel function.
Laguerre's polynomials satisfy the recurrence relations
and
in particular
and
or
moreover
They can be used to derive the four 3-point-rules
combined they give this additional, useful recurrence relations
A somewhat curious identity, valid for integer i and n, is
it may be used to derive the partial fraction decomposition
Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
moreover, this following equation holds
which generalizes with Cauchy's formula to
The derivate with respect to the second variable has the surprising form
The generalized associated Laguerre polynomials obey the differential equation
which may be compared with the equation obeyed by the k-th derivative of the ordinary Laguerre polynomial,
where for this equation only.
This points to a special case () of the formula above: for integer the generalized polynomial may be written , the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
In Sturm-Liouville form the differential equation is
which shows that is an eigenvector for the eigenvalue .
The associated Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e −x:[5]
which follows from
If denoted the Gamma distribution then the orthogonality relation can be written as
The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)
recursively
Moreover,
in the associated L2[0, ∞)-space.
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
Then
The series converges in the associated Hilbert space , iff
A related series expansion is
in particular
which follows from
Secondly,
a consequence derived from
for .
Monomials are represented as
binomials have the parametrization
This leads directly to
and, even more generally,
For a non-negative integer this simplifies to
for to
Jacobi's theta function has the representation
the Bessel function can be expressed (using an arbitrarily chosen parameter ) as
and thus
Gamma function has the parametrization
the lower incomplete Gamma function has the representations
and
The upper incomplete gamma function then is
where denotes the hypergeometric function.
Erdélyi gives the following two multiplication theorems [6]
The polynomials may be expressed in terms of a contour integral
where the contour circles the origin once in a counterclockwise direction.
The generalized Laguerre polynomials are related to the Hermite polynomials:
and
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.
The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as
where is the Pochhammer symbol (which in this case represents the rising factorial).
In terms of modified Bessel functions (Bessel polynomials) these following relations hold:
or further elaborated