On the characteristic function of the asymmetric Student's $t$-distribution and an integral involving the sine function

TL;DR

This paper derives new closed-form formulas for the characteristic function of the asymmetric Student's t-distribution and an integral involving sine, expanding analytical tools for these functions and their limits.

Contribution

It introduces novel closed-form expressions for the characteristic function of the asymmetric Student's t-distribution and a related sine integral, linking them to exponential integral functions.

Findings

New formula for the characteristic function of the asymmetric Student's t-distribution.

Closed-form expression for a sine integral involving exponential integral functions.

Limit formula involving special functions as the degrees of freedom approach integers.

Abstract

We obtain a new closed-form formula for the characteristic function of the asymmetric Student's $t$-distribution. As part of our analysis, we derive a new closed-form formula for the integral $\int_0^\infty \sin(ax)/(b^2+x^2)^n\,\mathrm{d}x$, for $a,b>0$, $n\in\mathbb{Z}^+$, expressed in terms of the exponential integral function. As a consequence of our integral formula, we deduce a closed-form formula for the limit $\lim_{\nu\rightarrow n} \{I_{\nu-1/2}(x)-\mathbf{L}_{1/2-\nu}(x)\}/\sin(\pi\nu)$, for $n\in\mathbb{Z}^+$, $x>0$.