  In this work, the authors provide closed forms and recurrence expressions for
computing the $k$th power of the formal power series, some of them in terms of
a determinant of some matrices. As a consequence, we obtain the reciprocal of
the unit of any formal power series. We apply these results to the generalized
Bernoulli numbers and Bernoulli numbers, we derive new closed-form expressions
and some recursive relations of these famous numbers. In addition, we present
several identities in determinant form. Using these results, an elegant
generalization of a well known identity of Euler is presented. We also note
some connections between the Stirling numbers of the second kind and the
generalized Bernoulli numbers.
