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In this work, the authors provide closed forms and recurrence expressions forcomputing the $k$th power of the formal power series, some of them in terms ofa determinant of some matrices. As a consequence, we obtain the reciprocal ofthe unit of any formal power series. We apply these results to the generalizedBernoulli numbers and Bernoulli numbers, we derive new closed-form expressionsand some recursive relations of these famous numbers. In addition, we presentseveral identities in determinant form. Using these results, an elegantgeneralization of a well known identity of Euler is presented. We also notesome connections between the Stirling numbers of the second kind and thegeneralized Bernoulli numbers.

Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers