Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

• Autorzy: Chen Xue-Yan , Wu Lan , Lim Dongkyu , Qi Feng

Identyfikatory

DOI

10.1515/dema-2022-0166

• Języki publikacji: EN

Abstrakty

EN

In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

Słowa kluczowe

EN

Bernoulli number; identity; product; cosecant; cotangent; hyperbolic cosecant; hyperbolic cotangent; series expansion; central factorial number of the second kind; closed-form formula

PL

liczba Bernoulliego; tożsamość; produkt; funkcje trygonometryczne; funkcja hiperboliczna; silnia; formuła zamknięta

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 822--830
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Chen Xue-Yan

• College of Engineering, Key Laboratory of Intelligent Manufacturing Technology, Inner Mongolia Minzu University, Tongliao 028000, Inner Mongolia, China

autor

Wu Lan

• College of Engineering, Inner Mongolia Minzu University, Tongliao 028000, Inner Mongolia, China

autor

Lim Dongkyu

autor

Qi Feng

• Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454003, China
• Independent Researcher, Dallas, TX 75252-8024, USA

Bibliografia

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• [3] F. Qi, On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers, Contrib. Discrete Math. 18 (2023), no. 1, in press.
• [4] Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 3, Paper No. 135, 12 pages, DOI: https://doi.org/10.1007/s13398-021-01071-x.
• [5] P. L. Butzer, M. Schmidt, E. L. Stark, and L. Vogt, Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim. 10 (1989), no. 5–6, 419–488, DOI: https://doi.org/10.1080/01630568908816313.
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• [7] J. Riordan, Combinatorial Identities, Reprint of the 1968 original, Robert E. Krieger Publishing Co., Huntington, N.Y., 1979.
• [8] F. Qi and B.-N. Guo, Relations among Bell polynomials, central factorial numbers, and central Bell polynomials, Math. Sci. Appl. E-Notes 7 (2019), no. 2, 191–194, DOI: https://doi.org/10.36753/mathenot.566448.
• [9] F. Qi, G.-S. Wu, and B.-N. Guo, An alternative proof of a closed formula for central factorial numbers of the second kind, Turk. J. Anal. Number Theory 7 (2019), no. 2, 56–58, DOI: https://doi.org/10.12691/tjant-7-2-5.
• [10] R. E. Haddad, A generalization of multiple zeta values. Part 2: Multiple sums, Notes Number Theory Discrete Math. 28 (2022), no. 2, 200–233, DOI: https://doi.org/10.7546/nntdm.2022.28.2.200-233.
• [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015, DOI: https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
• [12] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.
• [13] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Eds), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010, http://dlmf.nist.gov/.
• [14] F. Qi and P. Taylor, Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind, arXiv (2022), available online at https://arxiv.org/abs/2204.05612v4.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).