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

Said Zriaa; Mohammed Mouçouf

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arXiv · 2305.06072

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

Said Zriaa·Mohammed Mouçouf

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Abstract

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.

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Said Zriaa, Mohammed Mouçouf. 2023-05-10. Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers. https://arxiv.org/abs/2305.06072

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