bernoulli number
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2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minghui You

AbstractBy introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.


2021 ◽  
Vol 19 (1) ◽  
pp. 569-582
Author(s):  
Minghui You ◽  
Wei Song ◽  
Xiaoyu Wang

Abstract 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. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.


2020 ◽  
Vol 57 (8) ◽  
pp. 989-999
Author(s):  
Daniel Suescún-Díaz ◽  
Geraldyne Ule-Duque ◽  
Freddy Humberto Escobar

2019 ◽  
Vol 16 (05) ◽  
pp. 981-1003
Author(s):  
Hui-Qin Cao ◽  
Yuri Matiyasevich ◽  
Zhi-Wei Sun

In this paper, we establish some congruences involving the Apéry numbers [Formula: see text]. For example, we show that [Formula: see text] for any positive integer [Formula: see text], and [Formula: see text] for any prime [Formula: see text], where [Formula: see text] is the [Formula: see text]th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Aṕery numbers, [Formula: see text] and [Formula: see text].


2019 ◽  
Vol 2 (1) ◽  
pp. 43-47 ◽  
Author(s):  
Ikhsan Maulidi ◽  
Vina Apriliani ◽  
Muhamad Syazali

In this article, we study about the value of Riemann Zeta Function for even numbers using Bernoulli number. First, we give some basic theory about Bernoulli number and Riemann Zeta function. The method that used in this research was literature study. From our analysis, we have a theorem to evaluate the value of Riemann Zeta function for the even numbers with its proving.


2018 ◽  
Vol 16 (1) ◽  
pp. 1048-1060 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian

AbstractIn this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have$$\begin{array}{} \displaystyle {\it \Gamma} \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1} {\frac{a_{n}}{x^{2n-1}}}+R_{n}\left( x\right) \right) \end{array}$$with$$\begin{array}{} \displaystyle \left\vert R_{n}\left( x\right) \right\vert \leq \frac{\left\vert B_{2n}\right\vert }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{array}$$for all x > 0, where an has a closed-form expression, B2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.


2017 ◽  
Vol 13 (08) ◽  
pp. 1983-1993 ◽  
Author(s):  
Guo-Shuai Mao

In this paper, we show that for any prime [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Bernoulli polynomial of degree [Formula: see text]. And we prove that [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the [Formula: see text]th Bernoulli number. This confirms several conjectures of Z.-W. Sun.


2017 ◽  
Vol 13 (03) ◽  
pp. 705-716 ◽  
Author(s):  
Michael E. Hoffman

For [Formula: see text], let [Formula: see text] be the sum of all multiple zeta values with even arguments whose weight is [Formula: see text] and whose depth is [Formula: see text]. Of course [Formula: see text] is the value [Formula: see text] of the Riemann zeta function at [Formula: see text], and it is well known that [Formula: see text]. Recently Shen and Cai gave formulas for [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. We give two formulas for [Formula: see text], both valid for arbitrary [Formula: see text], one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers [Formula: see text] and for the analogous numbers [Formula: see text] defined using multiple zeta-star values of even arguments.


2015 ◽  
Vol 58 (3) ◽  
pp. 637-651 ◽  
Author(s):  
William Y. C. Chen ◽  
Jeremy J. F. Guo ◽  
Larry X. W. Wang

AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.


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