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 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, use the term for all polynomials in the Askey scheme. 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 (also called standardization) is detailed below. Jacobi polynomials For 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 The Jacobi polynomials with are called the Gegenbauer polynomials (with parameter ) For , these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1): For , one obtains the Chebyshev polynomials (of the second and first kind, respectively).