On the characteristic function of the asymmetric Student's $t$-distribution and an integral involving the sine function
Robert E. Gaunt
TL;DR
This work addresses the lack of a correct closed-form characteristic function for the asymmetric Student's $t$-distribution by deriving a new CF formula in terms of $K$, $I$, and $\mathbf{L}$ functions along with the exponential integral $\mathrm{Ei}$. It also provides a general closed-form for the integral $\int_0^\infty \frac{\sin(ax)}{(b^2+x^2)^n}\,dx$ for all $n\in\mathbb{Z}^+$ and a corollary for the limit $\lim_{\nu\to n} \frac{I_{\nu-1/2}(x)-\mathbf{L}_{1/2-\nu}(x)}{\sin(\pi\nu)}$, connecting to known Bessel/Struve identities. The CF reduces to the classical Student's $t$ CF in the symmetric case $\alpha=\tfrac{1}{2}$ and $\nu_1=\nu_2=\nu$, and the integral formula generalizes and unifies prior results, with extensions to a location-scale family for practical applications in statistics and finance.
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 $\lim_{ν\rightarrow n} \{I_{ν-1/2}(x)-\mathbf{L}_{1/2-ν}(x)\}/\sin(πν)$, for $n\in\mathbb{Z}^+$, $x>0$.