  We introduce q-frequently hypercyclic operators and derive a sufficient
criterion for a continuous operator to be q-frequently hypercyclic on a locally
convex space. Applications are given to obtain q-frequently hypercyclic
operators with respect to the norm-, F-norm- and weak*- topologies. Finally,
the frequent hypercyclicity of the non-convolution operator $T_\mu$ defined by
$T_\mu(f)(z) = f'(\mu z)$, $\mu\ge1$ on the space $H(\mathbb{C})$ of entire
functions equipped with the compact-open topology is shown.
