Bernoulli Numbers Research Articles

The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution

AbstractThe factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n \gt 0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int _0^1 b_n(x) dx = 0$ . We offer a related characterization: $b_1(x) = x - 1/2$ and $({-}1)^{n-1} b_n(x)$ for $n \gt 0$ is the $n$ -fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta $(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$ . This result has a novel combinatorial analog, the Bernoulli clock: mark the hours of a $2 n$ hour clock by a uniformly random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$ , meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$ , then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$ , and so on. Starting from hour $0 = 2n$ , move clockwise to the first hour marked $1$ , continue clockwise to the first hour marked $2$ , and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$ . We show that for each positive integer $n$ , the event $( I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$ , where $n! b_n(0) = B_n(0)$ is the $n$ th Bernoulli number. For $ 1 \le k \le 2 n$ , the difference $\delta _n(k)\,:\!=\, 1/(2n) -{\mathbb{P}}( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $\delta _n( 2 n + 1 - k) = ({-}1)^n \delta _n(k)$ , which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials $b_n(1-x) = ({-}1)^n b_n(x)$ .

Extension of Hoffman’s Combinatorial Identity via Specific Zeta-Like Series

The goal of the paper is twofold. First, we present an analytic method leading to a class of combinatorial identities with Bernoulli, Euler and Catalan numbers based on considering specific multiple zeta-like series and infinite products. The developed method allows us to naturally extend Hoffman’s combinatorial identity that led to the famous evaluation of the multiple zeta value ζ({2}k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta (\\{2\\}_k)$$\\end{document} in 1992. Second, we present new evaluations of two multiple zeta-like series with their consequences to combinatorial identities, and, as a by-product of our technical considerations, we establish two combinatorial identities with a trinomial coefficient and Stirling numbers respectively.

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

Exact solutions of burst pressure for thick-walled cylinders in power-law strain hardening steels

Exact solutions of burst pressure for defect-free, thick-walled cylindrical pressure vessels with capped ends are developed using the flow theory of plasticity in terms of the Tresca, von Mises, and Zhu-Leis yield criteria. The material is assumed to obey a power-law strain hardening rule, and the finite strain theory is adopted to describe large plastic deformation. On this basis, internal pressure is obtained as a complex polynomial function of effective strains on the inside and outside surfaces of the pressure vessel with the use of the Bernoulli numbers. At burst failure, the effective strains and stresses on the inside and outside surfaces are first determined, and then three flow solutions of burst pressure are obtained as a power series function of the diameter ratio (Do/Di), strain hardening exponent (n), and ultimate tensile stress (UTS). The power series solution is confirmed to agree well with the exact flow solution of burst pressure in a closed-form for each yield criterion, and the results show that the von Mises flow solution is an upper bound prediction, the Tresca flow solution is a lower bound prediction, and the Zhu-Leis flow solution is an intermediate prediction of burst pressure. Two sets of full-scale burst test data are then utilized to evaluate and validate the proposed flow solutions of burst pressure for both thin-walled pipes and thick-walled cylindrical pressure vessels.

Two Forms for Maclaurin Power Series Expansion of Logarithmic Expression Involving Tangent Function

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers.

Some New Optimal Bounds for Wallis Ratio

Wallis ratio can be expressed as an asymptotic expansion using Stirling series and Bernoulli numbers. We prove the general inequalities for Wallis ratio for arbitrary number of terms in the asymptotic expansion. We show that the coefficients in the asymptotic expansion are the best possible.

Starlike Functions Associated with Bernoulli’s Numbers of Second Kind

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let φBS(ξ)=ξeξ−12=∑n=0∞ξnBn2n!, where the coefficients of Bn2 are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions 𝟊 such that ξ𝟊′(ξ)𝟊(ξ)≺φBS(ξ). We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class.

Explicit relations on the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials and numbers

The main aim of this paper is to introduce and investigate the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle and operational methods. We give explicit relations and some identities for the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials.

Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients.

Exact solution of burst pressure for thick-walled pipes using the flow theory of plasticity

This paper develops an exact solution of burst pressure for defect-free, thick-walled pipes using the flow theory of plasticity in terms of the average shear stress yield criterion (or Zhu-Leis yield criterion). The pipe steel is assumed to obey the power-law strain hardening rule, and large plastic deformation is described by the finite strain theory. On this basis, internal pressure is obtained as a power series function of the effective strains on the inside and outside surfaces of the thick-walled pipe with use of the second Bernoulli numbers. At burst failure, the Zhu-Leis flow solution of burst pressure is determined as a power series solution, and the burst effective strains, the burst effective stresses, and the burst pressure are functions of the diameter ratio (Do / Di), strain hardening exponent (n), and ultimate tensile strength (UTS). Similarly, the Tresca and von Mises flow solutions of burst pressure are also determined as a power series solution in terms of the Tresca and von Mises yield criteria. A general closed-from exact solution of burst pressure is then proposed for the three yield criteria, and the results showed that the proposed exact solution matches well with the power series solution for each yield criterion. Moreover, the von Mises flow solution is an upper bound prediction, the Tresca flow solution is a lower bound prediction, and the Zhu-Leis flow solution is an intermediate prediction that agrees well with the finite element analysis results of burst pressure for thick-walled pipes. Two datasets of full-scale burst tests are then utilized to evaluate and validate the proposed flow solutions of burst pressure for both thin and thick-walled pipes.

Hadamard product of series with special numbers

We evaluate in closed form a large class of power series where the coefficients are products of special numbers, or products of special numbers and Laguerre polynomials. A number of examples is presented involving harmonic, hyperharmonic, Fibonacci, Lucas, Bernoulli, Euler, and derangement numbers. We also obtain a new convolution formula for the Laguerre polynomials.

Divided powers in the Witt ring of symmetric bilinear forms

The Witt ring of quadratic forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are linear combinations of exterior powers. We find the explicit formula for the divided powers as a linear combination of exterior powers. The coefficients involve the ``tangent numbers'', related to Bernoulli numbers. The divided powers on the Witt ring give another construction of the divided powers on Milnor K-theory modulo 2.

Some identities involving Bernoulli, Euler and degenerate Bernoulli numbers and their applications

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable.

Prime divisors of ℓ-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level ℓ

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1−ℓn)Bn, with Bn the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…,Gp−3, and ℓ-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ.

Reciprocal Symmetry via Inverse Series Pairs

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers.

Canonical equivariant cohomology classes generating zeta values of totally real fields

It is known that the special values at nonpositive integers of a Dirichlet L L -function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L L -functions of totally real fields. Hecke L L -functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.

Beyond the Basel Problem: Part II

Summary In the 18th century, Euler evaluated the sum in terms of the Bernoulli number for any positive integer k. In this article we evaluate the sum where a and b are integers satisfying and .

On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli–Fibonacci and generalized (p,q)-Bernoulli–Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli–Fibonacci and generalized bivariate (p,q)-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials.

Random recursive hypergraphs

Random recursive hypergraphs (RRHs) grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat expressions via harmonic numbers, Bernoulli numbers, Eulerian numbers, and Stirling numbers of the first kind. Natural deformations of RRHs give rise to fascinating models of growing random hypergraphs.

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.