Editing Classical orthogonal polynomials (section)

{{Use American English|date = March 2019}}
{{Short description|Type of 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]]<ref>See {{harvtxt|Suetin|2001}}</ref>).

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 fraction]]s to solve the [[moment problem]] by [[Pafnuty Chebyshev|P. L. Chebyshev]] and then [[Andrey Markov|A.A. Markov]] and [[Thomas Joannes Stieltjes|T.J. Stieltjes]] led to the general notion of orthogonal polynomials.

For given [[polynomial]]s <math>Q, L: \R \to \R</math> and <math>\forall\,n \in \N_0</math> the classical orthogonal polynomials <math>f_n:\R \to \R</math> are characterized by being solutions of the differential equation 
:<math>Q(x) \, f_n^{\prime\prime} +  L(x)\,f_n^{\prime} + \lambda_n  f_n = 0</math>
with to be determined constants <math>\lambda_n \in \R</math>.  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 <math>\lambda_n</math>.

There are several more general definitions of orthogonal classical polynomials; for example, {{harvtxt|Andrews|Askey|1985}} use the term for all polynomials in the [[Askey scheme]].