  A generalization of the canonical coherent states of a quantum harmonic
oscillator has been performed by requiring the conditions of normalizability,
continuity in the label and resolution of the identity operator with a positive
weight function. Relying on this approach, in the present scenario coherent
states are generalized over the canonical or finite dimensional Fock space of
the harmonic oscillator. A class of generalized coherent states is determined
such that the distribution of the number of excitations departs from the
Poisson statistics according to combinations of stretched exponential decays,
power laws and logarithmic forms. The analysis of the Mandel parameter shows
that these generalized coherent states exhibit (non-classical) sub-Poissonian
or super-Poissonian statistics of the number of excitations for small values of
the label, according to determined properties. The statistics is uniquely
sub-Poissonian for large values of the label. As particular cases, truncated
Wright generalized coherent states exhibit uniquely non-classical properties,
differently from the truncated Mittag-Leffler generalized coherent states.
