Approximating the coefficients of the Bessel functions

Abstract

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 $|\hat{I}¸N|\to \infty$ regime for the type A and D root systems and over the $|\hat{I}{¸}_{0}N|\to \infty ,\frac{\hat{I}{¸}_1}{\hat{I}{¸}_{0}N}\to c\in \mathbb{C}$ regime for the type BC root system. We also establish equivalent conditions over the $\hat{I}¸N\to c\in \mathbb{C}$ regime for the type A and D root systems and over the $\hat{I}{¸}_{0}N\to c_0\in \mathbb{C},\frac{\hat{I}{¸}_1}{\hat{I}{¸}_{0}N}\to c_1\in \mathbb{C}$ 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.