The Zappa-Szép product of a Fell bundle and a groupoid

We define the Zappa-Szép product of a Fell bundle by a groupoid, which turns out to be a Fell bundle over the Zappa-Szép product of the underlying groupoids. Under certain assumptions, every Fell bundle over the Zappa-Szép product of groupoids arises in this manner. We then study the representation associated with the Zappa-Szép product Fell bundle and show its relation to covariant representations. Finally, we study the associated universal C⁎-algebra, which turns out to be a C⁎-blend, generalizing an earlier result about the Zappa-Szép product of groupoid C⁎-algebras. In the case of discrete groups, the universal C⁎-algebra of a Fell bundle embeds injectively inside the universal C⁎-algebra of the Zappa-Szép product Fell bundle.