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.
Sometimes the name Laguerre polynomials is used for solutions of
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 Gaussian quadrature to numerically compute integrals of the form
These polynomials, usually denoted "L"0, "L"1, ..., 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 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 .
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.)
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:
Furthermore,
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.
For arbitrary real α the polynomial solutions of the differential equation
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 of the generalized Laguerre polynomials:
The Rodrigues formula for them is
The generating function for them 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:
Laguerre's polynomials satisfy the recurrence relations
in particular
and
or
moreover
They can be used to derive the four 3-point-rules
combined they give this additional, useful recurrence relations
Since formula_42 is a monic polynomial of degree formula_43 in formula_44,
there is the partial fraction decomposition
The second equality follows by the following identity, valid for integer "i" and and immediate from the expression of formula_42 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 () 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.
Moreover, the following equation holds:
which generalizes with Cauchy's formula to
The derivative with respect to the second variable has the form, 
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 "k"th derivative of the ordinary Laguerre polynomial,
where formula_55 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
If formula_59 denotes the Gamma distribution then the orthogonality relation can be written as
The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)
recursively
Moreover,
Turán's inequalities can be derived here, which is
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
Series expansions.
Let a function have the (formal) series expansion
Then
The series converges in the associated Hilbert space if and only if
Further examples of expansions.
Monomials are represented as
while binomials have the parametrization
This leads directly to
for the exponential function. The incomplete gamma function has the representation
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 "H""n"("x") are the Hermite polynomials based on the weighting function exp(−"x"2), 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, as
where formula_76 is the Pochhammer symbol (which in this case represents the rising factorial).
Hardy–Hille formula.
The generalized Laguerre polynomials satisfy the Hardy–Hille formula
where the series on the left converges for formula_78 and formula_79. Using the identity
(see generalized hypergeometric function), this can also be written as
This formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.