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. The Wikipedia article Rodrigues' formula has a proof that the polynomials obtained from the Rodrigues' formula obey a differential equation of this form and also derives.
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 satisfy
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 following 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 as 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:
  • The solutions are a series of polynomials P0, P1, P2,..., each Pn having degree n, and corresponding to a number λn.
  • The interval of orthogonality is bounded by whatever roots Q has.
  • The root of L is inside the interval of orthogonality.
  • Letting, the polynomials are orthogonal under the weight function
  • W has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
  • W gives a finite inner product to any polynomials.
  • W can be made to be greater than 0 in the interval.
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:
  • For fixed r, the polynomial sequence P, P, P,... are orthogonal, weighted by.
  • P is proportional to
  • P is a solution of, where λr is the same function as λn, that is,
  • P is a solution of
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
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".
  • Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is , and has Q = 1 − x2. They can then be standardized into the Jacobi polynomials. There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev.
  • Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is, and has Q = x. They can then be standardized into the Associated Laguerre polynomials. The plain Laguerre polynomials are a subclass of these.
  • Every Hermite-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is, and has Q = 1 and L = 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 , 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.