The central result of this paper consists in proving that all stable densities are bell-shaped (i.e. its $k$th derivative has exactly $k$ zeros and they are simple) thereby generalizing the well-known property of the normal distribution and the associated Hermite polynomials.
Wolfgang Gawronski. "On the Bell-Shape of Stable Densities." Ann. Probab. 12 (1) 230 - 242, February, 1984. https://doi.org/10.1214/aop/1176993386