On three summation equations for functions that are holomorphic in the plane with a cut along a polygonal line

We study three four-element summation equations in the class of functions that are holomorphic outside a polygonal line and vanish at infinity. The polygonal line is part of the boundary of a unit square. We seek a solution in the form of a Cauchy-type integral with unknown density satisfying some additional conditions. The regularization of the equation on the polygonal line is achieved by introducing an involutive piecewise-linear shift that reverses the orientation of the line. We rely on the contraction mapping method in a Banach space to prove that the resulting Fredholm equation of the second kind is solvable. Finally, we give the conditions for the equivalence of the regularization and consider some applications to interpolation problems for entire functions.