  New asymptotic approximations of the non-central $t$ distribution are given,
a generalization of the Student's $t$ distribution. Using new integral
representations, we give new asymptotic expansions for large values of the
noncentrality parameter but also for large values of the degrees of freedom
parameter. In some case we accept more than one large parameter. These results
are in terms of elementary functions, but also in terms of the complementary
error function and the incomplete gamma function. A number of numerical tests
demonstrate the performance of the asymptotic approximations.
