In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the 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 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, Andrews & Askey (1985) use the term for all polynomials in the Askey scheme.
Read more about Classical Orthogonal Polynomials: Definition, Differential Equation, Derivation From Differential Equation, Characterizations of Classical Orthogonal Polynomials, Table of Classical Orthogonal Polynomials
Famous quotes containing the word classical:
“Et in Arcadia ego.
[I too am in Arcadia.]”
—Anonymous, Anonymous.
Tomb inscription, appearing in classical paintings by Guercino and Poussin, among others. The words probably mean that even the most ideal earthly lives are mortal. Arcadia, a mountainous region in the central Peloponnese, Greece, was the rustic abode of Pan, depicted in literature and art as a land of innocence and ease, and was the title of Sir Philip Sidney’s pastoral romance (1590)