The Bessel function expression of characteristic function

TL;DR

This paper derives unified expressions for the characteristic functions of a wide class of elliptical and related distributions, including important special cases, using Bessel functions and hypergeometric series.

Contribution

It provides simple closed-form characteristic functions for key distributions like Student-t, Cauchy, and stable, using Bessel and hypergeometric functions.

Findings

Unified expressions for elliptical distribution characteristic functions.

Closed-form formulas for multivariate Student-t, Cauchy, and stable distributions.

Expressions involve Bessel functions and hypergeometric series.

Abstract

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.