Bernoulli Numbers Research Articles

A symbolic approach to multiple Hurwitz zeta values at non-positive integers

Abstract In this article, we give another method to calculate the values of multiple Hurwitz zeta function at non-positive integers by means of Raabe’s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.

Heat coefficients for magnetic Laplacians on the complex projective space P n(ℂ)

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν .

Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers

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 \[\displaystyle \sum_{k=0}^{n}\frac{(-1)^{k}h_{k}}{4^{k}} {{2k} \choose {k}}G_{n-k}=\frac{(-1)^{n-1}}{2^{2n-1}}{{2n-2} \choose {n-1}}\] and \[\displaystyle \sum_{k=1}^{\infty}\frac{h_{k}}{k^{2}(2k-1)4^{k}} {{2k} \choose {k}}=2\pi +3\zeta(2)\log 2-3\zeta(2)-\frac{7}{2}\zeta(3),\] where h_k=H_{2k}-\dfrac{1}{2}H_{k}, G_k are Bernoulli numbers of the second kind, and \zeta is the Riemann zeta function. We also give an alternate proof of the series representations for the constants \log (2 \pi) and \gamma given by Blagouchine and Coppo.

New inequalities of Huygens-type involving tangent and sine functions

Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.

A Quadratic Relation in the Bernoulli Numbers

We present a proof of an identity involving the Bernoulli numbers. This identity has been proved, over the centuries, by many different methods. Our method uses a technique from over a millennium ago for finding polynomial expressions for sums of powers of integers.

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

<abstract><p>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.</p></abstract>

Sharp inequalities related to the Adamovic-Mitrinovic, Cusa, Wilker and Huygens results

In this paper, we establish sharp inequalities for trigonometric functions. For example, we consider the Wilker inequality and prove that for 0 < x < ?/2 and n ? 1, 2 + (?n?1 j=2 dj+1x2j+ ?nx2n) x3 tan x < (sin x/x)2 + tan x/x < 2 + (?n?1 j=3 dj+1x2j+ Dnx2n) x3 tan x with the best possible constants ?n = dn and Dn = 2?6 ? 168?4 + 15120/945?4 (2/?) 2n ? ?n?1 j=2 dj+1 (2/?/)2n?2j , where dk = 22k+2 ((4k + 6) |B2k+2| + (?1)k+1)/(2k + 3)! and Bk are the Bernoulli numbers (k ? N0 := N? {0}). This improves and generalizes the results given by Mortici, Nenezic and Malesevic.

COEFFICIENT BOUNDS FOR <i>Q</i>-STARLIKE FUNCTIONS ASSOCIATED WITH <i>Q</i>-BERNOULLI NUMBERS

This paper's main goal is to introduce and study a subclass $\mathcal{S}% ^{\ast }(b,q)$ of $q$-starlike functions in the unit disk defined by the $q$-Bernoulli numbers. We determine the coefficient bounds, the upper bounds for the Fekete-Szegö functional, and the second Hankel determinant for this subclass.

Discrete quasiprobability distributions involving Bernoulli polynomials

<abstract><p>The aim of this short paper is to present a new family of discrete densities with two parameters based on Bernoulli numbers and polynomials. We use the properties of such numbers in order to compute the first moments and the density of a finite sum of such independent variables.</p></abstract>

Congruences involving generalized Catalan numbers and Bernoulli numbers

<abstract><p>In this paper, we establish some congruences mod $ p^3 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l} $, where $ p > 3 $ is a prime number and $ B_{p, k} $ are generalized Catalan numbers. We also establish some congruences mod $ p^2 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2} $, where $ m, l_1, l_2, d $ are positive integers and $ 1\leq d\leq p-1 $.</p></abstract>

Rounding error analysis of linear recurrences using generating series

We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.

Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ \mathbb{Z}_p $

<abstract><p>The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $ p $-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.</p></abstract>

A note on degenerate multi-poly-Bernoulli numbers and polynomials

In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate some properties for those numbers and polynomials. In addition, we give some identities and relations for the degenerate multi-poly- Bernoulli numbers and polynomials.

IDENTITIES INVOLVING GENERALIZED BERNOULLI NUMBERS AND PARTIAL BELL POLYNOMIALS WITH THEIR APPLICATIONS

In the present paper, we show that the partial Bell polynomials allow for obtaining identities involving the generalized Bernoulli numbers. Then, on applying these identities we derive different generating and bilateral generating functions.

A new combinatorial identity for Bernoulli numbers and its application in Ramanujan’s expansion of harmonic numbers

We establish a new combinatorial identity related to the well-known Bernoulli numbers, which generalizes the result due to Feng and Wang. By means of the identity, we find a recursive formula for successively determining the coefficients of Ramanujan?s asymptotic expansion for the generalized harmonic numbers

The denominators of the Bernoulli numbers

We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.

A new approach to fully degenerate Bernoulli numbers and polynomials

In this paper, we consider the doubly indexed sequence a(r) ? (n,m), (n,m ? 0), defined by a recurrence relation and an initial sequence a(r) ? (0,m), (m ? 0). We derive with the help of some differential operator an explicit expression for a(r) ? (n, 0), in term of the degenerate r-Stirling numbers of the second kind and the initial sequence. We observe that a(r) ? (n, 0) = ?n,?(r), for a(r) ? (0,m) = 1/m+1 , and a(r) ? (n, 0) = En,?(r), for a(r) ? (0,m) = (1/2)m . Here ?n,?(x) and En,?(x) are the fully degenerate Bernoulli polynomials and the degenerate Euler polynomials, respectively.

Generating functions for reciprocal Catalan-type sums: approach to linear differentiation equation and ($p$-adic) integral equations

This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the $p$-adic Volkenborn integral to the Catalan-type polynomials, some combinatorial sums and identities involving the Bernoulli numbers, the Stirling numbers and the Catalan-type numbers are derived.

Additional Fibonacci-Bernoulli relations

We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.

Periodic solutions for a higher-order p-Laplacian neutral differential equation with multiple deviating arguments

In this article, we consider the following high-order p-Laplacian neutral differential equation with multiple deviating arguments:$$(\varphi_{p}(x(t)-cx(t-r))^{(m)}(t)))^{(m)}= f(x(t))x'(t)+g(t,x(t),x(t-\tau_{1}(t)),...,x(t-\tau_{k}(t)))+e(t).$$By appling the continuation theorem, theory of Fourier series, Bernoulli numbers theory and some analytic techniques, sufficient conditions for the existence of periodic solutions are established.