On a Difference Equation Generated by Two “Close” Squares

Abstract

We consider a set of two squares constructed for the primitive periods 1 and i and having four vertices on one straight line. In a neighborhood of this set, we study a four-element difference equation with constant coefficients. The linear shifts of this equation are the generating transformations of the corresponding doubly periodic group and their inverse transformations. The solution is sought in the class of functions that are analytic outside this set and vanish at infinity. We give some applications to the moment problem for entire functions of exponential type.