Approximating the coefficients of the Bessel functions

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 $|\theta N| \rightarrow\infty$ regime for the type A and D root systems and over the $|\theta_0 N|\rightarrow \infty, \frac{\theta_1}{\theta_0 N}\rightarrow c\in\mathbb{C}$ regime for the type BC root system. We also establish equivalent conditions over the $\theta N \rightarrow c\in\mathbb{C}$ regime for the type A and D root systems and over the $\theta_0 N\rightarrow c_0\in\mathbb{C}, \frac{\theta_1}{\theta_0 N}\rightarrow c_1\in\mathbb{C}$ 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.