Higher Order Dualities over Global Function Fields and Weighted M\"{o}bius Sums over $\mathbb{F}_q{[T]}$

Alladi's duality identities (1977) provide a fundamental relation between the smallest and the $k$-th largest prime factors of integers. In this paper, we establish these dualities in the setting of global function fields, extending a result of Duan, Wang, and Yi (2021) to higher orders. We apply this to study a function field analogue of the sum $\sum \mu(n)\omega(n)/n$, when restricted to integers whose smallest prime factor lies in an arbitrary subset of primes possessing a natural density. These results demonstrate how the second-order duality identity governs the asymptotic behaviour of these weighted M\"{o}bius sums in the function field setting.