  Some special functions are particularly relevant in applied probability and
statistics. For example, the incomplete beta function is the cumulative central
beta distribution. In this paper, we consider the inversion of the central
Student's-$t$ distribution which is a particular case of the central beta
distribution. The inversion of this distribution functions is useful in
hypothesis testing as well as for generating random samples distributed
according to the corresponding probability density function. A new asymptotic
representation in terms of the complementary error function, will be one of the
important ingredients in our analysis. As we will show, this asymptotic
representation is also useful in the computation of the distribution function.
We illustrate the performance of all the obtained approximations with numerical
examples.
