Bernoulli Numbers Research Articles

Congruence relations of Ankeny–Artin–Chowla type for real quadratic fields

In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of [Formula: see text] for positive square-free integers [Formula: see text]. Many of the ideas present in their paper can be seen as the precursors to the now developed theory of cyclotomic fields. Curiously, little attention has been paid to the cases of [Formula: see text] in the literature. In this work, we show that the congruences of the type proven by Ankeny, Artin, and Chowla can be seen as a special case of a more general methodology using Kubota–Leopoldt [Formula: see text]-adic [Formula: see text]-functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in [Formula: see text] by relating these values to a well known expression for [Formula: see text]. We conclude with a discussion of known counterexamples to the so-called Composite Ankeny–Artin–Chowla conjecture and relate these to special dihedral extensions of [Formula: see text].

A new generalization of Ibn al-Haytham recursive formula for sums of integer powers

For integers $n,k \geq 1$, let $S_k(n)$ denote the sum of $k$-th powers of the first $n$ positive integers. In this paper we derive a family of recursive formulas for $S_k(n)$, generalizing the classical recursive formula discovered by Ibn al-Haytham. As a result, we provide a number of new recursive formulas expressing $S_k(n)$ in terms of the lower-order power sums $S_0(n), S_1(n),\ldots, S_{k-1}(n)$. Also, we present several recursive formulas for $S_k(n)$ involving the Bernoulli numbers, the harmonic numbers, the Stirling numbers of the second kind, and a type of generalized harmonic numbers.

Some identities related to degenerate Bernoulli and degenerate Euler polynomials

ABSTRACT The aim of this paper is to study degenerate Bernoulli and degenerate Euler polynomials and numbers and their higher-order analogues. We express the degenerate Euler polynomials in terms of the degenerate Bernoulli polynomials and vice versa. We prove the distribution formulas for degenerate Bernoulli and degenerate Euler polynomials. We obtain some identities among the higher-order degenerate Bernoulli and higher-order degenerate Euler polynomials. We express the higher-order degenerate Bernoulli polynomials in x + y as a linear combination of the degenerate Euler polynomials in y . We get certain identities involving the degenerate r -Stirling numbers of the second and the binomial coefficients.

Probabilistic poly-Bernoulli numbers

ABSTRACT Assume that is Y a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to study probabilistic poly-Bernoulli numbers associated with Y, as probabilistic extensions of poly-Bernoulli numbers. We derive explicit expressions, some related identities and a symmetric relation for those numbers. We also investigate explicit expressions for the modified probabilisitc Bernoulli numbers associated with Y, which are slightly different from probabilisitic Bernoulli numbers associated with Y. As special cases of Y, we treat the Poisson, gamma and Bernoulli random variables.

Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Abstract In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

Low-energy limit of N-photon amplitudes in a constant field: Part II

We employ the worldline formalism to derive a series representation of the low-energy limit of the N-photon amplitude in a constant background field for both scalar and spinor QED. The amplitudes are then written in terms of a single proper-time integral. The above-mentioned series representation terminates when considering a constant crossed field. This allows us to obtain even more compact expressions for these particular amplitudes for which the result of the proper-time integral, for fixed parameters, takes the form of a factorial function. In addition, we derive all helicity components of these amplitudes and express them explicitly in terms of Bernoulli numbers and spinor products.

On normalized tails of series expansion of generating function of Bernoulli numbers

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials.

Emergent family of Tsallis entropies from the q-deformed combinatorics

Emergent family of Tsallis entropies from the q-deformed combinatorics

Formulas for Bernoulli Numbers and Polynomials

Formulas for Bernoulli Numbers and Polynomials

Normalized Gompertz wavelets and their applications

In this paper we define, using the Stirling numbers of the second kind, the Gompertz wavelets and show their basic properties. In particular, we prove that the admissibility condition holds for them. We also compute the normalizing factors in the space of the square-integrable functions [Formula: see text] and present an explicit formula for them in terms of the Bernoulli numbers. Then, after implementing the second-order Gompertz wavelets in the Matlab Wavelet Toolbox, we perform numerical experiments demonstrating the practical applicability and effectiveness of the wavelets.

Generalized circular impact time guidance

Generalized circular impact time guidance

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤp

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind.

Bernoulli numbers in embedding constants for Sobolev spaces with diverse boundary conditions

The Sobolev spaces W 2 n [ 0 ; 1 ] W^n_2[0;1] are treated with five different boundary conditions (periodic, antiperiodic, of even and odd orders, and of even-odd order). Sharp estimates for the derivatives of order k = 0 , 1 , … , n − 1 k=0,1,\dots , n-1 are obtained, sharp constants for the embedding of W 2 n [ 0 ; 1 ] W^n_2[0;1] into W ∞ k [ 0 ; 1 ] W^k_\infty [0;1] are found, it is shown that they are rationally expressed in terms of Bernoulli numbers and, therefore, are rational. The exact embedding constants can also be expressed in terms of the Riemann ζ \zeta -function. The reproducing kernels of these spaces are calculated.

Bernoulli numbers with level 2

Bernoulli numbers with level 2

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on Zp

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials.

Formulas for Bernoulli and Euler Numbers and Polynomials with the aid of Applications Operators and Volkenborn Integral

Not only are there operators for studying properties for special numbers and polynomials, but the Volkenborn integral has an equally powerful applications. The aim of this article is to derive new formulas by applying operators and ($p$-adic)the Volkenborn integral to certain families polynomial, especially the Euler polynomials. These formulas include the Stirling numbers, array polynomials, the Fubini type polynomials, and the Bernoulli and Euler numbers and polynomials.

Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-Volkenborn integral including the Volkenborn integral and p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some various identities, formulas and relations related to several combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. By using generating functions, we give relations between the Lah numbers, the Bernoulli numbers, the Euler numbers and the Laguerre polynomials. We also give further comments and remarks on these functions, numbers and integral formulas related to q-type operators potentially used to solve problems in the areas such as physics, quantum mechanics, quantum systems and the others. In addition, we provide some tables containing some of the p-adic integral formulas obtained in this paper.

New moment formulas for moments and characteristic function of the geometric distribution in terms of Apostol–Bernoulli polynomials and numbers

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol–Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler's identity, we give relations among this characteristic function, the Apostol–Bernoulli polynomials and numbers, and also trigonometric functions including and . A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol–Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas.

NOVEL WAY OF DETERMINING SUM OF KTH POWERS OF NATURAL NUMBERS

Since ancient times, mathematicians across the world have worked on different methods to find the sum of powers of natural numbers. In this paper, we are going to present the relationship between sum of kth powers of natural numbers and sum of (k–1) th powers of natural numbers using the differential operator and associated recurrence relation. Interestingly, the Bernoulli numbers which occur frequently in mathematical analysis, play an important role in establishing this relationship.

On some coefficients of the Artin–Hasse series modulo a prime

On some coefficients of the Artin–Hasse series modulo a prime