Classical orthogonal polynomials


In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials.
They have many important applications in such areas as mathematical physics, 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 and the classical orthogonal polynomials are characterized by being solutions of the differential equation
with to be determined constants.
There are several more general definitions of orthogonal classical polynomials; for example, use the term for all polynomials in the Askey scheme.

Definition

In general, the orthogonal polynomials with respect to a weight
The relations above define up to multiplication by a number. Various normalisations are used to fix the constant, e.g.
The classical orthogonal polynomials correspond to the three families of weights:
The standard normalisation is detailed below.

Jacobi polynomials

For the Jacobi polynomials are given by the formula
They are normalised by
and satisfy the orthogonality condition
The Jacobi polynomials are solutions to the differential equation

Important special cases

The Jacobi polynomials with are called the Gegenbauer polynomials
For, these are called the Legendre polynomials :
For, one obtains the Chebyshev polynomials.

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 polynomial, and L is a given linear polynomial. The function f, and the constant λ, are to be found.
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 an eigenvector/eigenvalue problems: Letting D be the differential operator,, and changing the sign of λ, the problem is to find the eigenvectors f, and the
corresponding eigenvalues λ, such that f does not have singularities and D = λf.
The solutions of this differential equation have singularities unless λ takes on
specific values. There is a series of numbers λ0, λ1, λ2,... that led to a series of polynomial solutions P0, P1, P2,... if one of the following sets of conditions are met:
  1. 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.
  2. 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.
  3. 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:
Because of the constant of integration, the quantity R is determined only up to an arbitrary positive multiplicative constant. It will be used only in homogeneous differential equations
and in the definition of the weight function The tables below will give the "official" values of R and W.

Rodrigues' formula

Under the assumptions of the preceding section,
Pn is proportional to
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 ''λ''''n''

Under the assumptions of the preceding section, we have

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

Let
Then
Now multiply the differential equation
by S/Q, getting
or
But, so
or, letting u = Sy,

Formulas involving derivatives

Under the assumptions of the preceding section, let P denote the r-th derivative of Pn.
P is a polynomial of degree n − r. Then we have the following:
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:
The differential equation for a particular λ may be written
multiplying by yields
and reversing the subscripts yields
subtracting and integrating:
but it can be seen that
so that:
If the polynomials f are such that the term on the left is zero, and for, then the orthogonality relationship will hold:
for.

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".
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 , still have two parameters to be determined.
They are and in the Jacobi polynomials,
written. We have and
Both and are required to be greater than −1.
When and 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

When one sets the parameters and in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written, and defined as
We have and
The parameter is required to be greater than −1/2.
Ignoring the above considerations, the parameter is closely related to the derivatives of :
or, more generally:
All the other classical Jacobi-like polynomials are special cases of the Gegenbauer polynomials, obtained by choosing a value of 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, denoted
where and are integers with, are defined as
The m in parentheses 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, the sequence are orthogonal over , with weight 1.
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,
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 or .
There are also Chebyshev polynomials of the second kind, denoted
We have:
For further details, including the expressions for the first few
polynomials, see Chebyshev polynomials.

Laguerre polynomials

The most general Laguerre-like polynomials, after the domain has been shifted and scaled, are the Associated Laguerre polynomials, denoted. There is a parameter, which can be any real number strictly greater than −1. The parameter is put in parentheses to avoid confusion with an exponent. The plain Laguerre polynomials are simply the 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
The parameter is closely related to the derivatives of :
or, more generally:
Laguerre's equation can be manipulated into a form that is more useful in applications:
is a solution of
This can be further manipulated. When is an integer, and :
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.
Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of, 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
Because the multiplier is proportional to the square root of the weight function, these functions
are orthogonal over 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 of the continuous Fourier transform.
Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of instead of. 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, who showed that 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.