$f$-Frequently hypercyclic $C_0$-semigroups on complex sectors

We analyze $f$-frequently hypercyclic, $q$-frequently hypercyclic ($q>1$) and frequently hypercyclic $C_0$-semigroups ($q=1$) defined on complex sectors, working in the setting of separable infinite-dimensional Fr'echet spaces. Some structural results of ours are given for a general class of $\mathcal{F}$-frequently hypercyclic $C_0$-semigroups, as well. We investigate generalized frequently hypercyclic translation semigroups and generalized frequently hypercyclic semigroups induced by semiflows on weighted function spaces. Several illustrative examples are presented.