  For population genetics models with recombination, obtaining an exact,
analytic sampling distribution has remained a challenging open problem for
several decades. Recently, a new perspective based on asymptotic series has
been introduced to make progress on this problem. Specifically, closed-form
expressions have been derived for the first few terms in an asymptotic
expansion of the two-locus sampling distribution when the recombination rate
$\rho$ is moderate to large. In this paper, a new computational technique is
developed for finding the asymptotic expansion to an arbitrary order.
Computation in this new approach can be automated easily. Furthermore, it is
proved here that only a finite number of terms in the asymptotic expansion is
needed to recover (via the method of Pad\'{e} approximants) the exact two-locus
sampling distribution as an analytic function of $\rho$; this function is exact
for all values of $\rho\in[0,\infty)$. It is also shown that the new
computational framework presented here is flexible enough to incorporate
natural selection.
