  We consider here point processes $N^f(t)$, $t>0$, with independent increments
and integer-valued jumps whose distribution is expressed in terms of
Bern\v{s}tein functions $f$ with L\'evy measure $\nu$. We obtain the general
expression of the probability generating functions $G^f$ of $N^f$, the
equations governing the state probabilities $p_k^f$ of $N^f$, and their
corresponding explicit forms. We also give the distribution of the
first-passage times $T_k^f$ of $N^f$, and the related governing equation. We
study in detail the cases of the fractional Poisson process, the relativistic
Poisson process and the Gamma Poisson process whose state probabilities have
the form of a negative binomial. The distribution of the times $\tau_j^{l_j}$
of jumps with height $l_j$ ($\sum_{j=1}^rl_j = k$) under the condition $N(t) =
k$ for all these special processes is investigated in detail.
