Finite-dimensional representations of difference operators, and the identification of remarkable matrices

TL;DR

This paper introduces finite-dimensional matrix representations of difference operators acting on polynomials, enabling the transformation of difference equations into matrix equations and identifying remarkable matrices related to special polynomial zeros.

Contribution

The paper develops exact finite-dimensional matrix representations of difference operators and identifies notable matrices connected to polynomial zeros in the Askey schemes.

Findings

Matrices explicitly depend on N arbitrary numbers or polynomial zeros.

Transform difference equations into matrix-vector form.

Identification of matrices with Diophantine properties.

Abstract

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_{n}, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f(z) as follows: [f(z+a)-f(z)]/a respectively [f(qz)-f(z)]/[(q-1)z]. These representations are exact---in a sense explained in the paper---when the function f(z) is a polynomial in z of degree less than N. This formalism allows to transform difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers z_{n}, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character.