We determine equivalent conditions between the asymptotic coefficients of the Bessel generating functions of a sequence of probability measures and the asymptotic expected values of power sums when their inputs are sampled from these measures. We establish these conditions over the $|θN| \rightarrow\infty$ regime for the type A and D root systems and over the $|θ_0 N|\rightarrow \infty, \frac{θ_1}{θ_0 N}\rightarrow c\in\mathbb{C}$ regime for the type BC root system. We also establish equivalent conditions over the $θN \rightarrow c\in\mathbb{C}$ regime for the type A and D root systems and over the $θ_0 N\rightarrow c_0\in\mathbb{C}, \frac{θ_1}{θ_0 N}\rightarrow c_1\in\mathbb{C}$ regime for the type BC root system that generalize existing results. Furthermore, we determine the asymptotics of the coefficients of the Bessel functions over the regimes that we have mentioned.