Generating function for Hermitian and non-Hermitian models

It is well known that Hermitian and non-Hermitian models exhibit distinct physics and require different theoretical tools. In this work, we propose a unified generating-function framework for both classes with generic boundary conditions and local impurities. Within this framework, any finite lattice model can be mapped to a generating function of the form G(z)=P(z)/Q(z), where Q(z) and P(z) denote the bulk recurrence relation and boundary terms or impurities, respectively. The problem of solving for eigenstates reduces to a simple criterion based on the cancellation of zeros of Q(z) and P(z). Applying this method to the Hatano-Nelson (HN) model, we show how boundary conditions and impurities determine the location of the zeros, thereby demonstrating the boundary sensitivity of non-Hermitian systems. We further investigate topological edge states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model and identify its topological phase transition. Inspired by generating-function techniques widely used in discrete mathematics, particularly in the study of the Fibonacci sequence, our results establish a direct connection between non-Hermitian physics and recurrence relations, providing a new perspective for analyzing non-Hermitian systems and exploring their connections with discrete mathematical structures.