Some properties of the orthogonal polynomials of a discrete variable
This article presents a synthesis of certain properties of the most usual discrete orthogonal polynomials, analogous to those found in the mathematical literature for continuous orthogonal polynomials. The orthogonal polynomials of a discrete variable considered are the LEGENDRE, the LAGUERRE, the KRAWTCHOUCK, the HERMITE, the POISSON-CHARLIER. These polynomials possess the orthogonality respectively orthonormality property and are derived from a RODRIGUEZ formula. Some families of discrete orthogonal polynomials admit a transform matrix. Such a matrix, of dimensions (N,N), is formed by the evaluation of the N first polynomials of the family considered at x = 0,1,2,...,N-1. The computation of this matrix is based on the use of the shifted PASCAL matrix; this helps to unify the formulation of the orthogonal transforms. Some transform matrices are indicated, they have applications in signal processing, image processing, engineering and physics. It is shown that the discrete HERMITE polynomials are deduced from KRAWTCHOUK's polynomials by expressing both families of polynomials by means of hypergeometric functions.
Bibliographic Reference: EUR 17683 EN (1997) 26pp., FS, free of charge
Availability: Available from the Public Relations and Publications Unit, JRC Ispra, I-21020 Ispra (IT), Fax: +39-332-785818
Record Number: 199711410 / Last updated on: 1997-10-27
Original language: en
Available languages: en