In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Laguerre polynomials, named after
Edmond Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
(1834–1886), are solutions of Laguerre's equation:
which is a second-order
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
. 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
Nikolay Yakovlevich Sonin (Russian: Никола́й Я́ковлевич Со́нин, February 22, 1849 – February 27, 1915) was a Russian mathematician.
Biography
He was born in Tula and attended Lomonosov University, studying mathematic ...
).
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
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
to numerically compute integrals of the form
These polynomials, usually denoted , , …, are a
polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in en ...
which may be defined by the
Rodrigues formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...
,
reducing to the closed form of a following section.
They are
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
with respect to an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
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 are ...
,
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 sub ...
s in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the
Tricomi–Carlitz polynomials In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by and and , related to random walks on the positive integers.
They are given in terms of Laguerre polynomials In mathematics, ...
.
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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 quant ...
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
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
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 seri ...
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
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
for any :
The simple Laguerre polynomials are the special case of the generalized Laguerre polynomials:
The
Rodrigues formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...
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 seri ...
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 irregular ...
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 t ...
. 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 var ...
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 combin ...
, 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), 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
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
The second equality follows by the following identity, valid for integer ''i'' and 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 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 In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors.
If is ...
can be derived here, which is
The following
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
is needed in the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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 exponent ...
s are represented as
while
binomials have the parametrization
This leads directly to
for the exponential function. The
incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which ...
has the representation
In quantum mechanics
In quantum mechanics the Schrödinger equation for the
hydrogen-like atom
A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such as ...
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 additio ...
s
Relation to Hermite polynomials
The generalized Laguerre polynomials are related to the
Hermite polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well as ...
s:
where the are the
Hermite polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well as ...
s based on the weighting function , the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the
.
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 irregular ...
s, as
where
is the
Pochhammer symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
(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
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
), this can also be written as
This formula is a generalization of the
Mehler kernel The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
Mehler's formula
defined a function
and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) ba ...
for
Hermite polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well as ...
s, 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 irregular ...
.
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
*
*
{{Authority control
Polynomials
Orthogonal polynomials
Special hypergeometric functions