The Bessel function expression of characteristic function

The purpose of the present paper is to give unified expressions to the characteristic functions of all elliptical and related distributions. Those distributions including the multivariate elliptical symmetric distributions and some asymmetric distributions such as skew-elliptical distributions and their location-scale mixtures. In particular, we get simple closed form of characteristic functions for important cases such as the multivariate Student-$t$, Cauchy, logistic, Laplace, symmetric stable. The expressions of characteristic functions involve Bessel type functions or generalized hypergeometric series.