Generating functions with high-order poles are nearly polynomial
Robin Pemantle · Mathematics and Computer Science · 2000
Consider the problem of obtaining asymptotic information about a multidimensional arrayof numbers ar, given the generating function $$ F\left( z \right) = \sum olimits_{r} {{{a}_{r}}} {{z}^{r}} $$ When F is meromorphic, it is known how to obtain various asymptotic series for a r in decreasing powers of $$ \left| r \right|$$ .The purpose of this note is to show thatwhen the pole set of F has singularities of a certain kindthen there can be only finitely many terms in such an asymptotic series. As a consequence, in the presence of a singularity of this kind, the whole asymptotic series for ar, is an effectively computable object.