Bernoulli Numbers Research Articles

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

We revisit the derivation of a formula for the q-generalised multinomial coefficient rooted in the q-deformed algebra, a foundational framework in the study of nonextensive statistics. Previous approximate expressions in the literature diverge as q approaches 2 (or 0, depending on convention). In contrast, our derived formula provides an exact, smooth function for all real values of q, expressed as an infinite series expansion involving Tsallis entropies with sequential entropic indices, coupled with Bernoulli numbers. This formulation is achieved through the analytic continuation of the Riemann zeta function, stemming from the q-deformed factorials. Our formula thus offers a distinctive characterisation of Tsallis entropy within the q-deformed combinatorics. Throughout this exploration, we also highlight a symmetry within the q-deformed theory that links different values of the entropic parameter. Furthermore, we discuss extending our results to encompass more general, affinity-sensitive cases, building on the previously established framework of affinity-based extended entropy.

Formulas for Bernoulli Numbers and Polynomials

Formulas for Bernoulli Numbers and Polynomials

Generalized circular impact time guidance

To effectively control impact time and ensure terminal acceleration converges to zero under the look angle constraint, this paper proposes a generalized circular impact time guidance law, providing a deeper insight into circular guidance. The proposed guidance law is formulated based on a generalized circular guidance equation that offers an explicit solution for the time to go. The convergence of both miss-distance and terminal acceleration to zero is proven using Bernoulli number theory. Impact time control is achieved by employing a fixed-time convergent controller that guides the look angle to track desired values. Additionally, the look angle constraint is addressed by incorporating a well-designed function into the acceleration. The proposed guidance law does not require estimated time to go and small-angle assumptions for its implementation. By predicting the mean speed, the proposed guidance law remains valid for missiles with varying speeds. Furthermore, the planar results are extended to three-dimensional scenarios by adopting the concept of the engagement plane. Nonlinear simulations demonstrate the effectiveness, advantages, and robustness of the proposed guidance law in the presence of disturbances and autopilot lag.

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

Let p be an odd prime, and let ∑n=0∞anXn∈Fp[[X]] be the reduction modulo p of the Artin–Hasse exponential series. We obtain a polynomial expression for akp in terms of those arp with r<k, for even k<p3−1. A conjectural analogue covering the case of odd k<p can be stated in various polynomial forms, essentially in terms of the polynomial γ(X)=∑n=1p−2(Bn/n)Xp−n, where Bn denotes the nth Bernoulli number.We prove that γ(X) satisfies the functional equation γ(X−1)−γ(X)=£1(X)+Xp−1−wp−1 in Fp[X], where £1(X) and wp are the truncated logarithm and the Wilson quotient. This is an analogue modulo p of a functional equation, in Q[[X]], established by Zagier for the power series ∑n=1∞(Bn/n)Xn. The proof of our functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.

Curve fitting for seismic waves of earthquake with Hermite polynomials

We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil).

Explicit formulae for Bernoulli numbers

&lt;p&gt;By examining the connection coefficients, we systematically review and extend (with an extra integer parameter) several double sum expressions for the Bernoulli numbers. New summation formulae are also established explicitly.&lt;/p&gt;

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine

&lt;abstract&gt;&lt;p&gt;In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.&lt;/p&gt;&lt;/abstract&gt;

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 14 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers

In the paper, by virtue of Wronski's formula and Kaluza's theorem related to a power series and its reciprocal, by means of Cahill and Narayan's recursive relation, and with the aid of the logarithmic convexity of the sequence of the Bernoulli numbers, the author presents the signs of certain Toeplitz-Hessenberg determinants whose elements involve the Bernoulli numbers and combinatorial numbers. Moreover, with the help of a derivative formula for the ratio of two differentiable functions, the author provides an alternative proof of Wronski's formula.

On natural approaches related to classical trigonometric inequalities

In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for \(0 &lt; x &lt; \frac{\pi}{2}\) and any \(n \geq 5\) $$0 &lt; P_n(x)\ &lt;\ (\sin x)^2- x^3\cot x &lt; P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} - \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} $$ where \(P_n(x) = \sum_{3=k}^n a_k x^{2k+1}\) is a \(n\)-polynomial, with positive coefficients (\(k \geq 5\)), \(a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),\) \( B_{2k} \) are Bernoulli numbers. This improves a lot of lower bounds of \( \frac{\sin(x)}{x}\) and generalizes inequalities chains. Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function $$g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 - \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 1 $$