On a functional equation with holomorphic coefficients associated with a finite group

{We consider a convex pentagon $D$ that has a pair of parallel and equal sides without a common vertex. We study the linear difference equation associated with this polygon. The coefficients of the equation and the free term are holomorphic in $D$. The solution is sought in the class of functions holomorphic outside the "half" of the $\partial D$ boundary and vanishing at infinity. A method for its regularization is proposed and a condition for its equivalence is found. The solution is represented as a Cauchy-type integral with an unknown density. The principle of contraction mappings in a Banach space is essentially used. Applications to interpolation problems for entire functions of exponential type are indicated.