Bernoulli Polynomials Research Articles

A conjecture on Boros-Moll polynomials due to Ma, Qi, Yeh and Yeh

Gamma-positivity is one of the basic properties that may be possessed by polynomials with symmetric coefficients, which directly implies that they are unimodal. It originates from the study of Eulerian polynomials by Foata and Schützenberger. Then, the alternatingly gamma-positivity for symmetric polynomials was defined by Sagan and Tirrell. Later, Ma et al. further introduced the notions of bi-gamma-positive and alternatingly bi-gamma-positive for a polynomial f(x) which correspond to that both of the polynomials in the symmetric decomposition of f(x) are gamma-positive and alternatingly gamma-positive, respectively. In this paper we establish the alternatingly bi-gamma-positivity of the Boros–Moll polynomials by verifying both polynomials in the symmetric decomposition of their reciprocals are unimodal and alternatingly gamma-positive. It confirms a conjecture proposed by Ma, Qi, Yeh and Yeh.

Zeros distribution and interlacing property for certain polynomial sequences

Abstract In this article, we first prove that the Hankel determinant of order three of the polynomial sequence { P n ( x ) = ∑ k ≥ 0 P ( n , k ) x k } n ≥ 0 {\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P ( n , k ) P\left(n,k) satisfies the recurrence relation P ( n , k ) = ( a 1 n + a 2 ) P ( n − 1 , k ) + ( b 1 n + b 2 ) P ( n − 1 , k − 1 ) , P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P ( n , k ) = 0 P\left(n,k)=0 wherever k ∉ { 0 , 1 , … , n } . k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence ( U n ( x ) ) n ≥ 0 , {\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: U n ( x ) = ( α n x + β n ) U n − 1 ( x ) + ( u n x 2 + v n x ) U n − 1 ′ ( x ) {U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}^{2}+{v}_{n}x){U}_{n-1}^{^{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials.

Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

Write ζm(n), 1≤m≤n−1, for the negative zeros of the nth Bell polynomial, arranged in decreasing order. In this paper, we prove the following asymptotic: for every positive integer m we have limn→∞ζm(n)−mmm+1n−1=1.The approach used to find this asymptotic applies to many other significant families of polynomials. In particular, analogous asymptotics are also proved for the negative rightmost zeros of Eulerian polynomials, r-Bell polynomials, linear combinations of K consecutive Bell polynomials and many others.

On the $cd$-Index for Alternating Descents

In 2008, Chebikin considered the $cd$-index of $\mathfrak{S}_n$ with respect to the alternating descent statistic and asked for a combinatorial interpretation for its coefficients. In this paper, we provide an answer to Chebikin's open problem in terms of permutations without double descents and ending with an ascent, with respect to a new statistic defined on these permutations. Additionally, we demonstrate a $cd$-index approach to proving the gamma-expansions of alternating Eulerian polynomials. Furthermore, we offer a direct combinatorial interpretation for their gamma-coefficients.

Inclusive Subclasses of Bi-univalent Functions Specified by Euler Polynomials

Our research delineates novel subclasses of analytical functions using Euler polynomials. Afterwards, we estimate the Fekete -Szeg¨o functional problem and the Maclaurin coefficients for this subfamily, namely |a2| and |a3|. Additionally, several new results are shown to follow after specializing the parameters employed in our main results.

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.

Explicit construction of Hermitian Yang-Mills instantons on coset manifolds

In four dimensions, ’t Hooft symbols offer a compact and powerful framework for describing the self-dual structures fundamental to instanton physics. Extending this to six dimensions, the six-dimensional ’t Hooft symbols can be constructed using the isomorphism between the Lorentz group Spin(6) and the unitary group SU(4). We demonstrate that the six-dimensional self-dual structures governed by the Hermitian Yang-Mills equations can be elegantly organized using these generalized ’t Hooft symbols. We also present a systematic method for constructing Hermitian Yang-Mills instantons from spin connections on six-dimensional manifolds using the generalized ’t Hooft symbols. We provide a thorough analysis of the topological invariants such as instanton and Euler numbers.

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

A degenerate version of hypergeometric Bernoulli polynomials: announcement of results

Abstract This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function t m e λ x ( t ) e λ x ( t ) - ∑ l = 0 m - 1 ( 1 ) l , λ t l l ! = ∑ n = 0 ∞ B n , λ [ m - 1 ] ( x ) t n n ! , | t | < min { 2 π , 1 | λ | } , λ ∈ ℝ \ { 0 } . {{{t^m}e_\lambda ^x\left( t \right)} \over {e_\lambda ^x\left( t \right) - \sum\nolimits_{l = 0}^{m - 1} {\left( 1 \right)l,\lambda{{{t^l}} \over {l!}}} }} = \sum\limits_{n = 0}^{^\infty } {B_{n,\lambda }^{\left[ {m - 1} \right]}} \left( x \right){{{t^n}} \over {n!}},\,\,\,\,\left| t \right| < \min \left\{ {2\pi ,{1 \over {\left| \lambda \right|}}} \right\},\lambda \in \mathbb{R}\backslash \left\{ 0 \right\}. We deduce their associated summation formulas and their corresponding determinant form. Also we focus our attention on the zero distribution of such polynomials and perform some numerical illustrative examples, which allow us to compare the behavior of the zeros of degenerate hypergeometric Bernoulli polynomials with the zeros of their hypergeometric counterpart. Finally, using a monomiality principle approach we present a differential equation satisfied by these polynomials.

Bi-univalent characteristics for a particular class of Bazilevič functions generated by convolution using Bernoulli polynomials

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient | a 2 | and | a 3 | for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Formulas for Bernoulli Numbers and Polynomials

Formulas for Bernoulli Numbers and Polynomials

Numerical simulation for fractional optimal control problems via euler wavelets

Abstract This paper suggests employing Euler wavelets to provide a precise and effective computational approach for certain fractional optimal control problems. The primary objective of the approach is to transform the fractional optimal control problem defined by the dynamical system and performance index into systems of algebraic equations, which then can be readily solved using matrix techniques. Since Euler polynomials are employed to build Euler wavelets and since Euler polynomials have fewer terms than most of commonly used other polynomials to build wavelets, employing them for the numerical approach results in sparser operational matrices. The speed of the recommended numerical method is improved due to the fewer terms in Euler wavelet operational matrices. We obtain the corresponding systems of algebraic equations for state variable, control variable, and Lagrange multipliers (used to determine the essential conditions of optimality) by incorporating operational matrices of Euler wavelets. Subsequently, those systems of algebraic equations are solved to obtain the numerical solution. The suggested method’s efficiency is demonstrated using a few typical examples. The suggested procedure is accurate and efficient, according to the results.

More bijective combinatorics of weakly increasing trees

As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin–Ma–Ma–Zhou (2021). Various intriguing connections and bijections for weakly increasing trees have already been found and the purpose of this paper is to present yet more bijective combinatorics on this unified object. Two of our main contributions are•extension of an equidistribution result on plane trees due to Eu–Seo–Shin (2017), regarding levels and degrees of nodes, to weakly increasing trees;•a new interpretation of the multiset Schett polynomials in terms of odd left/right chains on weakly increasing binary trees. Interesting consequences are discussed, including new tree interpretations for the Jacobi elliptic functions and Euler numbers. Relevant enumerative results are also presented, involving recurrence relations, exponential generating functions and context-free grammars.

Information geometry and alpha-parallel prior of the beta-logistic distribution

The hyperbolic secant distribution has several generalizations with applications in, for example, finance. In this study, we explore the dual geometric structure of one such generalization: the beta-logistic distribution. Within this family, two special cases of random variables, as examples, are of particular interests: their moments, by some recent results, give the Bernoulli and Euler polynomials, which are important objects in many areas of mathematics. This current study also uncovers that the beta-logistic distribution admits a α-parallel prior for any real number α, that has the potential for application in geometric statistical inference.

Euler Polynomials and Bi-univalent Functions

Our research introduces new subclasses of analytical functions that are defined by Euler polynomials. We then proceed to estimate the Fekete-Szego functional problem and the Maclaurin coefficients for this specific subfamily, denoted as |a2| and |a3|. Furthermore, we demonstrate several new results that emerge when we specialize the parameters used in our main findings.

Generalization of Bernoulli polynomials to find optimal solution of fractional hematopoietic stem cells model

The study introduces a fractional mathematical model in the Caputo sense for hematopoietic stem cell-based therapy, utilizing generalized Bernoulli polynomials (GBPs) and operational matrices to solve a system of nonlinear equations. The significance of the study lies in the potential therapeutic applications of hematopoietic stem cells (HSCs), particularly in the context of HIV infection treatment, and the innovative use of GBPs and Lagrange multipliers in solving the fractional hematopoietic stem cells model (FHSCM). The aim of the study is to introduce an optimization algorithm for approximating the solution of the FHSCM using GBPs and Lagrange multipliers and to provide a comprehensive exploration of the mathematical techniques employed in this context. The research methodology involves formulating operational matrices for fractional derivatives of GBPs, conducting a convergence analysis of the proposed method, and demonstrating the accuracy of the method through numerical simulations. The major conclusion is the successful introduction of GBPs in the context of the FHSCM, featuring innovative control parameters and a novel optimization technique. The study also highlights the significance of the proposed method in providing accurate solutions for the FHSCM, thus contributing to the field of mathematical modeling in biological and medical research.

Bernoulli polynomials for a new subclass of Te-univalent functions

This paper introduces a novel subclass, denoted as Tσq,s(μ1;ν1,κ,x), of Te-univalent functions utilizing Bernoulli polynomials. The study investigates this subclass, establishing initial coefficient bounds for |a2|, |a3|, and the Fekete-Szegö inequality, namely |a3−ζa22|, are derived for this class. Additionally, several corollaries are provided to further elucidate the implications of the findings.

Hydrodynamics of a dual-orifice needle-free jet injector

In the recent literature, it has been shown that an enhanced, multi-faceted, balanced immune response can be generated using a combined intradermal (ID) and intramuscular (IM) delivery of vaccines, especially for novel DNA vaccines. However, needle-based injection for a combined ID/IM delivery may require separate injections, which may prove to be time consuming, and impractical for mass immunizations. In contrast, a novel multi-orifice jet injector could be used to deliver drugs at multiple penetration depths simultaneously, which may provide advantages such as ease of operation, elimination of sharps, and short injection timeframe (O (10 m)).Here, we consider a dual-orifice geometry with both a wide orifice (200 μm–400 μm) for IM drug delivery and a narrow orifice (100 μm) for ID drug delivery. Using numerical simulations, we found that, at a fixed upstream pressure, jet velocities through wide and narrow orifices do not vary significantly for low-viscosity fluids (≲4%). However, it was previously hypothesized that the jet power, Pj=π8ρdj2vj3, is a more appropriate parameter to characterize tissue penetration depth. Thus, the jet power could vary by a factor of ∼10, yielding different depths for the two orifices. Using non-dimensional analysis via Euler (Eu) and Reynolds (Re) numbers, we characterize the role of orifice geometry and driving pressure, to generate geometry-specific correlations in the Eu–Re parameter space to estimate the jet velocity and pressure losses. We also elucidate the orifice size ratios that are best suited for fractional doses of ∼100 μL to intradermal tissue from a total injection of 1 ml.Preliminary experiments to visualize the penetration of wide and narrow jet streams into gelatin and pork substrates showed that the wide jet stream leads to substantially greater penetration depth than the narrow jet stream. In summary, we provide the initial feasibility of simultaneous delivery of a drug to multiple penetration depths from the same injection cartridge using the needle-free jet injection technique.

Parameter identification of fractional-order systems with time delays based on a hybrid of orthonormal Bernoulli polynomials and block pulse functions

Parameter identification of fractional-order systems with time delays based on a hybrid of orthonormal Bernoulli polynomials and block pulse functions