Higher-order unification has been shown to be undecidable. Miller discovered
the pattern fragment and subsequently showed that higher-order pattern
unification is decidable and has most general unifiers. We extend the algorithm
to higher-order rational terms (a.k.a. regular B\"{o}hm trees, a form of cyclic
$\lambda$-terms) and show that pattern unification on higher-order rational
terms is decidable and has most general unifiers. We prove the soundness and
completeness of the algorithm