  The scale function holds significant importance within the fluctuation theory
of Levy processes, particularly in addressing exit problems. However, its
definition is established through the Laplace transform, thereby lacking
explicit representations in general. This paper introduces a novel series
representation for this scale function, employing Laguerre polynomials to
construct a uniformly convergent approximate sequence. Additionally, we derive
statistical inference based on specific discrete observations, presenting
estimators of scale functions that are asymptotically normal.
