Wikipedia / Classical orthogonal polynomials
Wikipedia / Classical orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials).
They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.
Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.
For given polynomials are characterized by being solutions of the differential equation
. The Wikipedia article Rodrigues' formula has a proof that the polynomials obtained from the Rodrigues' formula obey a differential equation.
There are several more general definitions of orthogonal classical polynomials; for example, use the term for all polynomials in the Askey scheme.
Definition
satisfy
up to multiplication by a number. Various normalisations are used to fix the constant, e.g.
The classical orthogonal polynomials correspond to the following three families of weights:
The standard normalisation (also called standardization) is detailed below.
Jacobi polynomials
the Jacobi polynomials are given by the formula
They are normalised (standardized) by
and satisfy the orthogonality condition
The Jacobi polynomials are solutions to the differential equation
Important special cases
are called the Gegenbauer polynomials)
, these are called the Legendre polynomials (for which the interval of orthogonality is [1,1] and the weight function is simply 1):
, one obtains the Chebyshev polynomials (of the second and first kind, respectively).
Hermite polynomials
The Hermite polynomials are defined by
They satisfy the orthogonality condition
and the differential equation
Laguerre polynomials
The generalised Laguerre polynomials are defined by
.)
They satisfy the orthogonality relation
and the differential equation
Differential equation
The classical orthogonal polynomials arise from a differential equation of the form
where Q is a given quadratic (at most) polynomial, and L is a given linear polynomial. The function f, and the constant λ, are to be found.
(Note that it makes sense for such an equation to have a polynomial solution.
Each term in the equation is a polynomial, and the degrees are consistent.)
This is a Sturm–Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of λ. They can be thought of as eigenvector/eigenvalue problems: Letting D be the differential operator, and changing the sign of λ, the problem is to find the eigenvectors (eigenfunctions) f, and the corresponding eigenvalues λ, such that f does not have singularities and D(f) = λf.
The solutions of this differential equation have singularities unless λ takes on specific values. There is a series of numbers 0, 1, 2, ... that led to a series of polynomial solutions P0, P1, P2, ... if one of the following sets of conditions are met:
Q is actually quadratic, L is linear, Q has two distinct real roots, the root of L lies strictly between the roots of Q, and the leading terms of Q and L have the same sign.
Q is not actually quadratic, but is linear, L is linear, the roots of Q and L are different, and the leading terms of Q and L have the same sign if the root of L is less than the root of Q, or vice versa.
Q is just a nonzero constant, L is linear, and the leading term of L has the opposite sign of Q.
These three cases lead to the Jacobi-like, Laguerre-like, and Hermite-like polynomials, respectively.
In each of these three cases, we have the following:
The solutions are a series of polynomials P0, P1, P2, ..., each Pn having degree n, and corresponding to a number n.
The interval of orthogonality is bounded by whatever roots Q has.
The root of L is inside the interval of orthogonality.
W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
W(x) gives a finite inner product to any polynomials.
W(x) can be made to be greater than 0 in the interval. (Negate the entire differential equation if necessary so that Q(x) > 0 inside the interval.)
Because of the constant of integration, the quantity R(x) is determined only up to an arbitrary positive multiplicative constant. It will be used only in homogeneous differential equations (where this doesn't matter) and in the definition of the weight function (which can also be indeterminate.) The tables below will give the "official" values of R(x) and W(x).
Rodrigues' formula
Under the assumptions of the preceding section, Pn(x
This is known as Rodrigues' formula, after Olinde Rodrigues. It is often written
where the numbers en depend on the standardization. The standard values of en will be given in the tables below.
The numbers λn
Under the assumptions of the preceding section, we have
(Since Q is quadratic and L are constants, so these are just numbers.)
Second form for the differential equation
Let
Then
Now multiply the differential equation
by R/Q, getting
or
This is the standard Sturm–Liouville form for the equation.
Third form for the differential equation
Then
Now multiply the differential equation
by S/Q, getting
or
, so
or, letting u = Sy,
Formulas involving derivatives
Under the assumptions of the preceding section, let P denote the r-th derivative of Pn. (We put the "r" in brackets to avoid confusion with an exponent.) P is a polynomial of degree nr. Then we have the following:
(orthogonality) For fixed r, the polynomial sequence P, P, P.
(generalized Rodrigues' formula) P
(differential equation) P, where r is the same function as n
(differential equation, second form) P
There are also some mixed recurrences. In each of these, the numbers a, b, and c depend on n and r, and are unrelated in the various formulas.
There are an enormous number of other formulas involving orthogonal polynomials in various ways. Here is a tiny sample of them, relating to the Chebyshev, associated Laguerre, and Hermite polynomials:
Orthogonality
The differential equation for a particular λ may be written (omitting explicit dependence on x)
yields
and reversing the subscripts yields
subtracting and integrating:
but it can be seen that
so that:
If the polynomials f, then the orthogonality relationship will hold:
.
Derivation from differential equation
All of the polynomial sequences arising from the differential equation above are equivalent, under scaling and/or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are exactly "classical orthogonal polynomials".
Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is [1,1], and has Q = 1x2. They can then be standardized into the Jacobi polynomials. There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev.
, and has Q = x. They can then be standardized into the Associated Laguerre polynomials. The plain Laguerre polynomials are a subclass of these.
, and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials.
Because all polynomial sequences arising from a differential equation in the manner described above are trivially equivalent to the classical polynomials, the actual classical polynomials are always used.
Jacobi polynomial
The Jacobi-like polynomials, once they have had their domain shifted and scaled so that the interval of orthogonality is [1, are required to be greater than 1. (This puts the root of L inside the interval of orthogonality.)
are not equal, these polynomials are not symmetrical about x = 0.
The differential equation
is Jacobi's equation.
For further details, see Jacobi polynomials.
Gegenbauer polynomials
in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical, and defined as
is required to be greater than1/2.
(Incidentally, the standardization given in the table below would make no sense for α = 0 and n instead of the value given in the table.)
:
or, more generally:
and choosing a standardization.
For further details, see Gegenbauer polynomials.
Legendre polynomials
The differential equation is
This is Legendre's equation.
The second form of the differential equation is:
The recurrence relation is
A mixed recurrence is
Rodrigues' formula is
For further details, see Legendre polynomials.
Associated Legendre polynomials
The Associated Legendre polynomials, are defined as
The m in parentheses (to avoid confusion with an exponent) is a parameter. The m in brackets denotes the m-th derivative of the Legendre polynomial.
These "polynomials" are misnamed—they are not polynomials when m is odd.
They have a recurrence relation:
For fixed m are orthogonal over [1,1], with weight1.
For given m are the solutions of
Chebyshev polynomials
The differential equation is
This is Chebyshev's equation.
The recurrence relation is
Rodrigues' formula is
These polynomials have the property that, in the interval of orthogonality,
(To prove it, use the recurrence formula.)
This means that all their local minima and maxima have values of 1 and +1, that is, the polynomials are "level". Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries.
Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0,1] or [2,2].
There are also Chebyshev polynomials of the second kind
We have:
For further details, including the expressions for the first few polynomials, see Chebyshev polynomials.
Laguerre polynomials
, which can be any real number strictly greater than version of these:
The differential equation is
This is Laguerre's equation.
The second form of the differential equation is
The recurrence relation is
Rodrigues' formula is
:
or, more generally:
Laguerre's equation can be manipulated into a form that is more useful in applications:
is a solution of
:
is a solution of
The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials:
This equation arises in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.
, than the definition used here.
For further details, including the expressions for the first few polynomials, see Laguerre polynomials.
Hermite polynomials
The differential equation is
This is Hermite's equation.
The second form of the differential equation is
The third form is
The recurrence relation is
Rodrigues' formula is
The first few Hermite polynomials are
One can define the associated Hermite functions
with no weight function.
The third form of the differential equation above, for the associated Hermite functions, is
The associated Hermite functions arise in many areas of mathematics and physics. In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator. They are also eigenfunctions (with eigenvalue (i n) of the continuous Fourier transform.
. If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by
For further details, see Hermite polynomials.
Characterizations of classical orthogonal polynomials
There are several conditions that single out the classical orthogonal polynomials from the others.
The first condition was found by Sonine (and later by Hahn), who showed that (up to linear changes of variable) the classical orthogonal polynomials are the only ones such that their derivatives are also orthogonal polynomials.
Bochner characterized classical orthogonal polynomials in terms of their recurrence relations.
Tricomi characterized classical orthogonal polynomials as those that have a certain analogue of the Rodrigues formula.
Table of classical orthogonal polynomials
The following table summarises the properties of the classical orthogonal polynomials.
See also
Askey scheme of hypergeometric orthogonal polynomials
Notes
References