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.
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
, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1): For
Such equations generally have singularities in their solution functions f except for particular values of λ.
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: These three cases lead to the Jacobi-like, Laguerre-like, and Hermite-like polynomials, respectively.
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.)
It is often written where the numbers en depend on the standardization.
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 (omitting explicit dependence on x) multiplying by
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.
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.
(Incidentally, the standardization given in the table below would make no sense for α = 0 and n ≠ 0, because it would set the polynomials to zero.
are special cases of the Gegenbauer polynomials, obtained by choosing a value of
, 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.
This means that all their local minima and maxima have values of −1 and +1, that is, the polynomials are "level".
Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0, 1] or [−2, 2].
The parameter is put in parentheses to avoid confusion with an exponent.
The second form of the differential equation is The recurrence relation is Rodrigues' formula is The parameter
Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of
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
In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator.
Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of
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.
The following table summarises the properties of the classical orthogonal polynomials.