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

Abstract

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers zn, 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 zn, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character.