Three identities and a determinantal formula for differences between Bernoulli polynomials and numbers

Jian Cao; José Luis López-Bonilla; Feng Qi; Jian Cao; José Luis López-Bonilla; Feng Qi

doi:10.3934/era.2024011

Electronic Research Archive

2024, Volume 32, Issue 1: 224-240. doi: 10.3934/era.2024011

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Research article

Three identities and a determinantal formula for differences between Bernoulli polynomials and numbers

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, Zhejiang, China
2.
ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX
3.
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
4.
School of Mathematics and Physics, Hulunbuir University, Inner Mongolia 021008, China
5.
Independent researcher, Dallas, TX 75252-8024, USA

Received: 14 February 2023 Revised: 11 August 2023 Accepted: 30 August 2023 Published: 21 December 2023

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.
- Bernoulli polynomial,
- Bernoulli number,
- difference,
- identity,
- inequality,
- determinant,
- partial Bell polynomial,
- determinantal formula
Citation: Jian Cao, José Luis López-Bonilla, Feng Qi. Three identities and a determinantal formula for differences between Bernoulli polynomials and numbers[J]. Electronic Research Archive, 2024, 32(1): 224-240. doi: 10.3934/era.2024011

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Abstract

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.

References

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