Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials

Aimin Xu; Aimin Xu

doi:10.3934/math.2025148

AIMS Mathematics

2025, Volume 10, Issue 2: 3197-3206. doi: 10.3934/math.2025148

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Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials

Aimin Xu ^,

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received: 23 December 2024 Revised: 14 February 2025 Accepted: 14 February 2025 Published: 20 February 2025
MSC : 05A19, 11B73, 11B83

By utilizing the generating function of higher-order Bernoulli polynomials, we uncover novel relationships that intertwine higher-order Bernoulli polynomials, higher-order Bernoulli numbers, Stirling numbers of the second kind, and central factorial numbers of the second kind. Leveraging these interconnections, we successfully rederive the identities formulated by Qi and Taylor, specifically those pertaining to Stirling numbers of the second kind and central factorial numbers of the second kind. Additionally, we derive series expansions for both positive integer and real powers of the sinc and sinhc functions.
- Bernoulli polynomial,
- Bernoulli number,
- series expansion,
- central factorial numbers of the second kind,
- Stirling numbers of the second kind
Citation: Aimin Xu. Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials[J]. AIMS Mathematics, 2025, 10(2): 3197-3206. doi: 10.3934/math.2025148

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Abstract

By utilizing the generating function of higher-order Bernoulli polynomials, we uncover novel relationships that intertwine higher-order Bernoulli polynomials, higher-order Bernoulli numbers, Stirling numbers of the second kind, and central factorial numbers of the second kind. Leveraging these interconnections, we successfully rederive the identities formulated by Qi and Taylor, specifically those pertaining to Stirling numbers of the second kind and central factorial numbers of the second kind. Additionally, we derive series expansions for both positive integer and real powers of the sinc and sinhc functions.

References

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