Eigenfunctions of Two-Scale Difference Equations and Appell Polynomials

Abstract

Both classical and distributional solutions of two-scale difference equations are interpreted as eigenfunctions, which are closely connected with Appell polynomials. Different generating functions are analyzed and the relations between them. Equivalent eigenfunctions as well as equivalent and minimal characteristic polynomials are defined and investigated in detail via the rational solution of a basic functional equation. Finally, reversed eigenfunctions are introduced and characterized.