On the difference equation associated with the doubly periodic group and its applications

Let $D$ be a rectangle. We consider a four-element linear difference equation defined on $D$. The shifts of this equation are the generating transformations of the corresponding doubly periodic group and their inverse transformations. We search for a solution in the class of functions that are holomorphic outside $D$ and vanish at infinity. Their boundary values satisfy a Hölder condition on any compact that does not contain the vertices. At the vertices, we allow, at most, logarithmic singularities. The independent term is holomorphic on $D$, and its boundary value satisfies a Hölder condition. The independent term may not be analytically continuable across an interval of the boundary, since the solution and the independent term belong to different classes of analytical functions. We regularize the difference equation and determine the conditions for the regularization to be equivalent. If the independent term is an odd function, then the problem is solvable. Additionally, we give some applications of the difference operator to interpolation problems for integer functions of exponential type and the construction of biorthogonally conjugated systems of analytical functions.