  Recently, the concept of generating function has been employed in one-loop
reduction. For one-loop integrals encompassing arbitrary tensor ranks and
higher-pole contributions, the generating function can be decomposed into a
tensor part and a higher-pole part. While the tensor component has been
thoroughly addressed in recent studies, there remains a lack of satisfactory
investigations regarding the higher-pole part. In this work, we completely
solve the problem. We first establish the partial differential equations
governing the higher-pole generating function. Based on these equations, we
derive an integration recursion relation and solve it iteratively. This
approach enables us to explore the analytical structure of higher-pole
reduction and provides a valuable tool for generating reduction coefficients
efficiently.
