It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole complex plane. The poles stem from the singularities of the rational function coefficients of the system. Just as for systems of differential equations, not all of these singularities necessarily lead to poles in a solution, as they might be what is called removable. In our work, we show how to detect and remove these singularities and further study the connection between poles of solutions, removable singularities and the extension of numerical sequences at these points.

众所周知，对于具有有理函数系数的线性差分方程的一阶系统，可以将在某个左半平面上的全纯解解析为在整个复平面上的亚纯解。极点来自系统有理函数系数的奇异性。就像微分方程组一样，并非所有这些奇异点都必然导致解中的极点，因为它们可能被称为可移动的。在我们的工作中，我们展示了如何检测和消除这些奇异点，并进一步研究了解的极点之间的联系，可移动奇异点和这些点处的数字序列的扩展。