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Several formulas for Bernoulli numbers and polynomials

  • * Corresponding author: Bijan Kumar Patel

    * Corresponding author: Bijan Kumar Patel 
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  • A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in the expansion $ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $, where $ a \neq 0, b $ are complex numbers, have studied in [16]. In this paper, we show that Bernoulli polynomials $ B_{p}(x) $ can be written in terms of the numbers $ S_{1,x}\left(p,k\right) $, and then use the known results for $ S_{1,x}\left(p,k\right) $ to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.

    Mathematics Subject Classification: Primary: 11B68, 11B73; Secondary: 11B83.


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