Frequently hypercyclic $C$-distribution semigroups and their generalizations

In this paper, we introduce the notions of $f$-frequent hypercyclicity and $\mathcal{F}$-hypercyclicity for $C$-distribution semigroups in separable Fr'echet spaces. We particularly analyze the classes of $q$-frequently hypercyclic $C$-distribution semigroups ($q\ge 1$) and frequently hypercyclic $C$-distribution semigroups, providing a great number of illustrative examples.