Discrete orthogonal transformations corresponding to the discrete polynomials of the Askey scheme

Abstract

This work presents discrete transforms that correspond to the discrete polynomials of the Askey Scheme. The Racah and Hahn transforms have been studied before, but the transforms corresponding to Krawtchouk, Gram, Meixner, and Charlier polynomials are new. All of the transforms (including those for Racah and Hahn polynomials) can be easily obtained by diagonalization of a symmetric tridiagonal matrix, and they have the property that UT U = UU T = I. For all of the transforms the classical and non-classical regions of the square screens are obtained. Discrete representations of Wigner d functions, Legendre polynomials, and anharmonic oscillator wave functions are found. A truncation procedure is used to obtain finite size square screens for Meixner and Charlier polynomials that have an infinite range for the x variable. Braun Potential functions provide important insight about the transforms and the accuracy of the truncation procedure. Some properties of symmetric tridiagonal matrices are used to find the desired transformations. © 2014 Springer International Publishing.