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 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 , , …, 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
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They a ...
,
The
rook polynomial
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any s ...
s 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
The Morse potential, named after physicist Philip M. Morse, is a convenient
interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quan ...
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.)
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 :
Furthermore,
In solution of some boundary value problems, the characteristic values can be useful:
The closed form is
The
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
for them likewise follows,
Polynomials of negative index can be expressed using the ones with positive index:
Relation to binary functions
There is a method to set Laguerre polynomials using functions which is related to binary expansion of
:
Here
with
.
Also
Here
is and
is a generalisation of .
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 :
The simple Laguerre polynomials are the special case of the generalized Laguerre polynomials:
The
Rodrigues formula for them is
The
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
for them is
Explicit examples and properties of the generalized Laguerre polynomials
* Laguerre functions are defined by
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregula ...
s and Kummer's transformation as
where
is a generalized
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...
. 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 representation, much like the closely related Hermite polynomials. Namely, let
and consider the differential operator
. Then
.
* The first few generalized Laguerre polynomials are:
* The
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
of the leading term is ;
* The
constant term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial
:x^2 + 2x + 3,\
the 3 is a constant term.
After like terms are comb ...
, which is the value at 0, is
* If is non-negative, then ''L''
''n''(''α'') has ''n''
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
, strictly positive
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
(notice that
is a
Sturm chain
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of ...
), which are all in the
interval
* The polynomials' asymptotic behaviour for large , but fixed and , is given by
and summarizing by
where
is the
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
.
As a contour integral
Given the generating function specified above, the polynomials may be expressed in terms of a
contour integral
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis ...
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
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 and immediate from the expression of
in terms of
Charlier polynomials In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier.
They are given in terms of the generalized hypergeometric function by
:C_n(x; \mu)= _2F_0(-n,-x;-; ...
:
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 times leads to
This points to a special case () of the formula above: for integer the generalized polynomial may be written
the shift by 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
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
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
over with respect to the measure with
weighting function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
:
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 In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states that
: \sum_^n \frac = \frac \frac
where ''f'j''(''x'') is the ''j''th term of a set of orthogonal polyn ...
)
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
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
,
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
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s 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 theorem
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
s
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 function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregula ...
s, as
where
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
and
. 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.
Physicist Scaling Convention
The generalized Laguerre polynomials are used to describe the quantum wavefunction for
hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
orbitals. In the introductory literature on this topic,
a different scaling is used for the generalized Laguerre polynomials than the scaling presented in this article. In the convention taken here, the generalized Laguerre polynomials can be expressed as
where
is the
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregula ...
.
In the physicist literature, such as
, the generalized Laguerre polynomials are instead defined as
The physicist version is related to the standard version by
There is yet another convention in use, though less frequently, in the physics literature. Under this convention the Laguerre polynomials are given by
See also
*
Angelescu polynomials
*
Bessel polynomials
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series
:y_n(x)=\sum_^n\frac\,\left(\frac ...
*
Denisyuk polynomials
*
Transverse mode
A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwav ...
, an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile.
Notes
References
*
* G. Szegő, ''Orthogonal polynomials'', 4th edition, ''Amer. Math. Soc. Colloq. Publ.'', vol. 23, Amer. Math. Soc., Providence, RI, 1975.
*
* B. Spain, M.G. Smith, ''Functions of mathematical physics'', Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
*
*
Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
,
Laguerre Polynomial, From MathWorld—A Wolfram Web Resource.
*
External links
*
*
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Polynomials
Orthogonal polynomials
Special hypergeometric functions