The scale function plays a significant role in the fluctuation theory of Lévy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, which generally lacks an explicit representation. This paper introduces a novel series representation for the scale function, utilizing Laguerre polynomials to construct a uniformly convergent approximation sequence. Additionally, we conduct statistical inference based on specific discrete observations and propose estimators for the scale function that are asymptotically normal.
