I'm working with an $n$-dimensional spherical t-distribution for which the density is defined as
$f(x) = \frac{\Gamma(\frac{n+\nu}{2})}{(\pi \sigma^2\nu)^{\frac n2}\Gamma(\frac \nu2)} \left( 1 + \frac{\|x|\|^2}{\nu \sigma^2} \right)^{- \frac{\nu+n}{2}},$
with $\sigma > 0$ a scale parameter and $\nu \geq 1$ the degrees of freedom.
I know that when we represent $X$ like $RU$ with $U$ the uniform distribution on the unit sphere of dimension $n$ and $R$ a 1-dimensional positive random variable independent of $U$, then $R^2/n \sim F(n, \nu)$.
I'm interested in the distribution of $X$ when projected onto a 1-dimensional subspace. Because of symmetry I can simply take the first component to project onto. Integrating over the other dimensions the density function should be obtained. I'm stuck however with the integral
$g(x_1) = \int_{-\infty}^\infty \dots \int_{-\infty}^\infty \left( 1 + \frac{x_1^2 + \dots + x_n^2}{\nu \sigma^2} \right)^{- \frac{\nu+n}{2}} dx_2 \dots dx_n.$
Any ideas about how to obtain the distribution of the projected variable?
