Several formulas for Bernoulli numbers and polynomials

Takao Komatsu; Bijan Kumar Patel; Claudio Pita-Ruiz

doi:10.3934/amc.2021006

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2023, Volume 17, Issue 3: 522-535. Doi: 10.3934/amc.2021006

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Several formulas for Bernoulli numbers and polynomials

1.
Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
2.
Department of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar 751024, India
3.
Universidad Panamericana. Facultad de Ingeniería., Augusto Rodin 498, Ciudad de México, 03920, México

^* Corresponding author: Bijan Kumar Patel
^* Corresponding author: Bijan Kumar Patel

Received: September 2020

Revised: January 2021

Early access: March 2021

Published: June 2023

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Abstract

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.

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

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