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 formula_1 and formula_2 the classical orthogonal polynomials formula_3 are characterized by being solutions of the differential equation 
with to be determined constants formula_5.
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 formula_6 with respect to a weight formula_7
The relations above define formula_6 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 (also called "standardization") is detailed below.
Jacobi polynomials.
For formula_12 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.
The Jacobi polynomials with formula_17 are called the Gegenbauer polynomials (with parameter formula_18)
For formula_19, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1):
For formula_21, 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.
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, formula_30, 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 "λ"0, "λ"1, "λ"2, ... that led to a series of polynomial solutions "P"0, "P"1, "P"2, ... if one of the following sets of conditions are met:
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"("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,
"P""n"("x") is proportional to formula_33
This is known as Rodrigues' formula, after Olinde Rodrigues. It is often written
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)
multiplying by formula_45 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 formula_51 for formula_52, then the orthogonality relationship will hold:
for formula_52.
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 [−1, 1], still have two parameters to be determined.
They are formula_61 and formula_62 in the Jacobi polynomials,
written formula_55. We have formula_64 and
formula_65.
Both formula_61 and formula_62 are required to be greater than −1.
When formula_61 and formula_62 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 formula_61 and formula_62 in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written formula_73, and defined as
We have formula_64 and
formula_76.
The parameter formula_61 is required to be greater than −1/2.
Ignoring the above considerations, the parameter formula_61 is closely related to the derivatives of formula_73:
or, more generally:
All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of formula_61 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
formula_89 where formula_90 and formula_91 are integers with formula_92, 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", the sequence formula_95 are orthogonal over [−1, 1], with weight 1.
For given "m", formula_89 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 [0, 1] or [−2, 2].
There are also Chebyshev polynomials of the second kind, denoted formula_102
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 (also called generalized Laguerre polynomials), denoted formula_57. There is a parameter formula_61, 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 formula_106 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 formula_61 is closely related to the derivatives of formula_57:
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 formula_118 is an integer, and formula_119:
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 formula_123, 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 formula_59 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.
Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of formula_137 instead of formula_138. 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.