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Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion
Mathematics, 2022 Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers.
Kwang-Wu Chen
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Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals
Mathematics, 2022 Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective.
Jing Zhang, Lixia Zhang, Caishi Wang
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Demonstratio Mathematica, 2022 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 ...
Chen Xue-Yan +3 more
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On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
Open Mathematics, 2022 In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-
You Minghui, Sun Xia, Fan Xiansheng
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On a new generalization of some Hilbert-type inequalities
Open Mathematics, 2021 In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established.
You Minghui, Song Wei, Wang Xiaoyu
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Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove Σⁿₖ₌ₒ((-1)ᵏhₖ/4ᵏ)$binom{2k}{k}$Gₙ₋ₖ = ((-1)ⁿ⁻¹/2^²ⁿ⁻¹)
Necdet Batır, Anthony Sofo
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Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function
Advances in Difference Equations, 2021 A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function.
Taekyun Kim +4 more
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Bernoulli Numbers and Solitons [PDF]
Journal of Nonlinear Mathematical Physics, 2005 We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx. $$ This formula is motivated by the results of Fairlie and Veselov, who discovered the relation of Bernoulli polynomials with soliton theory.
Grosset, Marie-Pierre +1 more
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Mathematics, 2020 In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots.
Jung Yoog Kang, Chen Seoung Ryoo
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Congruences among generalized Bernoulli numbers [PDF]
Acta Arithmetica, 1995 Let \(\chi\) denote a primitive quadratic character mod \(M\) (or the trivial character) and let \(d\) be a fundamental discriminant (or 1). Denote by \(\chi'\) the character mod \(M |d |\) induced by \(\chi\). The authors consider the generalized Bernoulli numbers \(B_{m, \chi'}\) and the corresponding Bernoulli polynomials \(B_{m, \chi'} (X)\) at \(X
Szmidt, J. +2 more
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