In this article, we study a class of lattice random variables in the domain of attraction of an $α$-stable random variable with index $α\in (0,2)$ which satisfy a truncated fractional Edgeworth expansion. Our results include studying the class of such fractional Edgeworth expansions under simple operations, providing concrete examples; sharp rates of convergence to an $α$-stable distribution in a local central limit theorem; Green's function expansions; and finally fluctuations of a class of discrete stochastic PDE's driven by the heavy-tailed random walks belonging to the class of fractional Edgeworth expansions.