We provide here some probabilistic interpretations of the generalized binomial distributions proposed by Gazeau et al. ["Generating functions for generalized binomial distributions," J. Math. Phys. 53, 103304 (2012)]. In the second part, we prove the "strong conjecture" expressed by Gazeau et al. about the coefficients of the Taylor expansion of the exponential of a polynomial. The proof relies mainly on properties of the Gould-Hopper polynomials. (C) 2013 AIP Publishing LLC.