Bernoulli Polynomials Research Articles

2D eπ-map for image encryption

A chaotic map generally employed to generate chaotic sequence is key element for an image encryption algorithm (IEA). In this study, an IEA using novel 2D chaotic map, which is based on Euler and Pi numbers so-called eπ-map, is presented. eπ-map exploits infinity diversity attribute of these numbers. Moreover, a diffusion operation referred to as “bit reversion” in which the bits of the pixels are symmetrically reverted is proposed for manipulating the pixel value. eπ-map is exhaustively examined through bifurcation and trajectory diagrams, Lyapunov exponent (LE), sample entropy (SE), permutation entropy (PE) and 0-1 test. The encryption performance of the IEA is then investigated across various cryptanalysis such as key-space, key sensitivity, histogram, information entropy, correlation coefficient, differential attack, cropping attack, noise attack and encryption execution time. Furthermore, the results are compared with the most recent literature. It is demonstrated that eπ-map has superior hyperchaotic performance in views of ergodicity, complexity and randomness. The IEA based on eπ-map and bit reversion is a secure and reliable algorithm thanks to its outperforming cryptanalysis results.

Construction of partially degenerate Laguerre–Bernoulli polynomials of the first kind

ABSTRACT In this paper, we introduce partially degenerate Laguerre–Bernoulli polynomials of the first kind and deduce some relevant properties by using a preliminary study of these polynomials. We derive some theorems on implicit summation formulae for partially degenerate Laguerre–Bernoulli polynomials of the first kind . Finally, we derive some symmetry identities for partially degenerate Laguerre–Bernoulli polynomials of the first kind.

Asymptotic Approximations of Apostol-Tangent Polynomials in Terms of Hyperbolic Functions

The tangent polynomials Tn (z) are generalization of tangent numbers or the Euler zigzag numbers Tn. In particular, Tn (0) = Tn. These polynomials are closely related to Bernoulli, Euler and Genocchi polynomials. One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostoltype polynomials. One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z, λ). When λ = 1, Tn (z, 1) = Tn(z). The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme. The same method was applied to the Genocchi polynomials by Corcino et al. The essential steps in applying the method are (1) to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula, and (2) to apply the saddle point method. It is found out that the method is applicable to Apostol-tangent polynomials. As a result, asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained. Moreover, higher-order Apostol-tangent polynomials are introduced. Using the same method, asymptotic approximation of higher-order Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained. It is important to note that the consideration of Apostol-type polynomials and higher order Apostoltype polynomials were not done by Lopez and Temme. This part is first done in this paper. The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n, λ and m.

A new generalization of the Genocchi numbers and its consequence on the Bernoulli polynomials

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by ${(B_n)}_{n \in \mathbb{N}}$ the sequence of the Bernoulli numbers and by ${(B_n(X))}_{n \in \mathbb{N}}$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $n$, the reciprocal polynomial of the polynomial $\big(B_n(X) - B_n\big)$ is integer-valued.

A matrix-collocation method for solutions of singularly perturbed differential equations via Euler polynomials

In this paper, a matrix-collocation method which uses the Euler polynomials is introduced to find the approximate solutions of singularly perturbed two-point boundary-value problems (BVPs). A system of algebraic equations is obtained by converting the boundary value problem with the aid of the collocation points. After this algebraic system, the coefficients of the approximate solution are determined. This error analysis includes two theorems which consist of an upper bound of errors and an error estimation technique. The present method and error analysis are applied to three numerical examples of singularly perturbed two-point BVPs. Numerical examples and comparisons with other methods are given. These applications and comparisons show that the method gives effective results.

Degenerate Poly‐Lah‐Bell Polynomials and Numbers

Many mathematicians studied “poly” as a generalization of the well‐known special polynomials such as Bernoulli polynomials, Euler polynomials, Cauchy polynomials, and Genocchi polynomials. In this paper, we define the degenerate poly‐Lah‐Bell polynomials arising from the degenerate polyexponential functions which are reduced to degenerate Lah‐Bell polynomials when k = 1. In particular, we call these polynomials the “poly‐Lah‐Bell polynomials” when λ⟶0. We give their explicit expression, Dobinski‐like formulas, and recurrence relation. In addition, we obtain various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly‐Bell polynomials, the degenerate poly‐Bernoulli numbers, and the degenerate poly‐Genocchi numbers.

Selection of Coherent and Concise Formulae on Bernoulli Polynomials-Numbers-Series and Power Sums-Faulhaber Problems

Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1st order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.

Representations of modified type 2 degenerate poly-Bernoulli polynomials

<abstract><p>Research on the degenerate versions of special polynomials provides a new area, introducing the $ \lambda $-analogue of special polynomials and numbers, such as $ \lambda $-Sheffer polynomials. In this paper, we propose a new variant of type 2 Bernoulli polynomials and numbers by modifying a generating function. Then we derive explicit expressions and their representations that provide connections among existing $ \lambda $-Sheffer polynomials. Also, we provide the explicit representations of the proposed polynomials in terms of the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials to confirm the presented identities.</p></abstract>

Identities on Changhee Polynomials Arising from λ‐Sheffer Sequences

In this paper, authors found a new and interesting identity between Changhee polynomials and some degenerate polynomials such as degenerate Bernoulli polynomials of the first and second kind, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Bell polynomials, degenerate Lah–Bell polynomials, and degenerate Frobenius–Euler polynomials and Mittag–Leffer polynomials by using λ‐Sheffer sequences and λ‐differential operators to find the coefficient polynomial when expressing the n‐th Changhee polynomials as a linear combination of those degenerate polynomials. In addition, authors derive the inversion formulas of these identities.

Several classical identities via Mellin’s transform

We present a summation rule using Mellin’s transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.

The extended Eulerian numbers over function fields

In this article, we introduce the extended Eulerian numbers for a large class of zeta functions, which includes the zeta functions associated to function fields, and to schemes over finite fields. This construction generalizes the extended Eulerian numbers defined by Carlitz. We give an asymptotic expansion for the summatory function associated to these numbers. Our main result generalizes the well known result on the asymptotic behavior of the extended Eulerian numbers associated to the Riemann zeta function.

WEIGHTED SYLVESTER SUMS ON THE FROBENIUS SET IN MORE VARIABLES

Let a1, a2, . . . , ak be positive integers with gcd(a1, a2, . . . , ak) = 1. Let NR = NR(a1, a2, . . . , ak) denote the set of positive integers non-representable in terms of a1, a2, . . . , ak. The largest non-representable integer max NR, the number of non-representable positive integers Σn∈NR 1, and the sum of non-representable positive integers Σn∈NR n have been widely studied for a long time as related to the famous Frobenius problem. In this paper, by using Eulerian numbers, we give formulas for the weighted sum Σn∈NR λnnμ, where μ is a non-negative integer and λ is a complex number. We also examine power sums of non-representable numbers and some formulas for three variables. Several examples illustrate and support our results.

Some Identities Involving the High-Order Cauchy Polynomials

In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

Asymptotic behavior of the Hurwitz–Lerch multiple zeta functions at non-positive integer points

We give a result on the asymptotic behavior of the Hurwitz–Lerch multiple zeta functions near non-positive integer points by using the Apostol–Bernoulli polynomials. From this result, we can evaluate limit values at non-positive integer points.

Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions

<abstract><p>We present a new type of degenerate poly-Bernoulli polynomials and numbers by modifying the polyexponential function in terms of the degenerate exponential functions and degenerate logarithm functions. Also, we introduce a new variation of the degenerate unipoly-Bernoulli polynomials by the similar modification. Based on these polynomials, we investigate some properties, new identities, and their relations to the known special functions and numbers such as the degenerate type 2-Bernoulli polynomials, the type 2 degenerate Euler polynomials, the degenerate Bernoulli polynomials and numbers, the degenerate Stirling numbers of the first kind, and $ \lambda $-falling factorial sequence. In addition, we compute some of the proposed polynomials and present their zeros and behaviors for different variables in specific cases.</p></abstract>

A note on the coefficients of power sums of arithmetic progressions

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind.

Cameron’s operator in terms of determinants and hypergeometric numbers

By studying Cameron’s operator in terms of determinants, two kinds of “integer” sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including s-step Fibonacci sequences. Another was the sequence of associated numbers, including Lamé sequences of higher order. By the classical Trudi’s formula and the inverse relation, more expressions were able to be obtained. These relations and identities can be extended to those of sequence of negative integers or rational numbers. As applications, we consider hypergeometric Bernoulli, Cauchy and Euler numbers with some modifications.

Formulae to Fubini Type Numbers emerge from Application of p-adic Integrals

The aim of this manuscript is to examine and survey various formulae for Fubini type numbers and polynomials with application of the p-adic integrals to some special polynomials. Relations and formulae related to the Fubini type numbers and polynomials, the Bernoulli numbers, the Euler numbers, Stirling type numbers, and combinatorial numbers are given. Moreover, by using generating functions with their functional equations, some new formulae including the Hermite polynomials, the Fubini type polynomials, and the Lah numbers are given. Finally, remarks on the results of this manuscript are presented.