On the Moyal Star Product of Resurgent Series

We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $i\hbar/2$ with coefficients in $C\{q,p\}$), which we show is stable under Moyal star product.