Bernoulli Polynomials Research Articles

Supercongruences involving Domb numbers and binary quadratic forms

In this paper, we prove two recently conjectured supercongruences (modulo p3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^3$$\\end{document}, where p is any prime greater than 3) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.

Descents on nonnesting multipermutations

Motivated by recent results on quasi-Stirling permutations, which are permutations of the multiset {1,1,2,2,…,n,n} that avoid the “crossing” patterns 1212 and 2121, we consider nonnesting permutations, defined as those that avoid the patterns 1221 and 2112 instead. We show that the polynomial giving the distribution of the number of descents on nonnesting permutations is a product of an Eulerian polynomial and a Narayana polynomial. It follows that, rather unexpectedly, this polynomial is palindromic. We provide bijective proofs of these facts by composing various transformations on Dyck paths, including the Lalanne–Kreweras involution.

Towards Solving Fractional Order Delay Variational Problems Using Euler Polynomial Operational Matrices

Towards Solving Fractional Order Delay Variational Problems Using Euler Polynomial Operational Matrices

New proofs of interlacing of zeros of Eulerian polynomials. III

New proofs of interlacing of zeros of Eulerian polynomials. III

Bernoulli polynomial based wavelets method for solving chaotic behaviour of financial model

This paper presents an algorithm for solving systems of non integer financial chaotic model. The Bernoulli wavelets function approximation applies to fractional order financial systems for the first time. The main objective of this process is to convert fractional differential equation systems into algebraic equation systems using wavelets approximation based on Bernoulli wavelets and their fractional integral operators. This fractional financial chaotic model that represents the effects of memory and chaos in a system. There are 3 compartments in this fractional order financial system: price indexes, interest rates, and investment demand. Analyses are depends on iteration method and Bernoulli wavelets method to determine the stability analysis, residual error and convergence analysis of the solutions. This study indicates that the presented approximate solutions fit exactly with the numerical solution and that the technique is accurate and effective. Our proposed model has been discussed for its uniqueness, boundedness, and non negativity. Its findings are extremely important for dealing with financial issues in the future.

Non-Orientable Lagrangian Fillings of Legendrian Knots

AbstractWe investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

Approximating a New Class of Stochastic Differential Equations via Operational Matrices of Bernoulli Polynomials

Approximating a New Class of Stochastic Differential Equations via Operational Matrices of Bernoulli Polynomials

On [formula omitted]th order Euler polynomials of degree [formula omitted] that are Eisenstein

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial Emp(mp)(x) is in Eisenstein form with respect to p if and only if p does not divide m(2m−1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials En(n)(x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q.

Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.

Characterization of the Bernoulli polynomials via the Raabe functional equation

Characterization of the Bernoulli polynomials via the Raabe functional equation

Carries and a map on the space of rational functions

A paper by Boros, Little, Moll, Mosteig, and Stanley relates properties of a map defined on the space of rational functions to Eulerian polynomials. We link their work to the carries Markov chain, giving a new proof and slight generalization of one of their results.

Thermohydraulic Analysis of Aqueous Dispersions Effectiveness with TiO<sub>2</sub> and Al<sub>2</sub>O<sub>3</sub> Nanoparticles in a U-Shaped Geothermal Probe

The paper deals with the influence of Al2O3 and TiO2 nanoparticle admixtures to the water coolant on the heat exchange and the pumping power of its single-phase flow in the U-shaped vertical collector of the heat pump geothermal probe. The heat carrier is aqueous dispersions with TiO2 and Al2O3 nanoparticles at volume concentrations of 0.1 %, 0.3 %, 0.6 %, 1.0 % and 1.3 %. Studies were performed for turbulent single-phase flow with a range of Reynolds numbers 11,500–67,000 and Euler numbers 48–75 and compared with similar results for a water coolant. The optimal volume fraction of nanoparticles in the coolant, at which nanofluids provide the lowest pumping capacity of the coolant compared to the base fluid, ensuring the required heat transfer coefficient at the level of 2230 W∙m-2∙K-1 is analyzed in the article.

Applications and Properties for Bivariate Bell-Based Frobenius-Type Eulerian Polynomials

In this study, we introduce sine and cosine Bell-based Frobenius-type Eulerian polynomials, and by presenting several relations and applications, we analyze certain properties. Our first step is to obtain diverse relations and formulas that cover summation formulas, addition formulas, relations with earlier polynomials in the literature, and differentiation rules. Finally, after determining the first few zero values of the Eulerian polynomials, we draw graphical representations of these zero values.

Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials.

Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

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.

The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

The hypersimplexΔk+1,n\Delta _{k+1,n}is the image of the positive GrassmannianGrk+1,n≥0Gr^{\geq 0}_{k+1,n}under the moment map. It is a polytope of dimensionn−1n-1inRn\mathbb {R}^n. Meanwhile, the amplituhedronAn,k,2(Z)\mathcal {A}_{n,k,2}(Z)is the projection of the positive GrassmannianGrk,n≥0Gr^{\geq 0}_{k,n}into the GrassmannianGrk,k+2Gr_{k,k+2}under a mapZ~\tilde {Z}induced by a positive matrixZ∈Matn,k+2>0Z\in Mat_{n,k+2}^{>0}. Introduced in the context ofscattering amplitudes, it is not a polytope, and has full dimension2k2kinsideGrk,k+2Gr_{k,k+2}. Nevertheless, there seem to be remarkable connections between these two objects viaT-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting outpositroid polytopes—images of positroid cells ofGrk+1,n≥0Gr^{\geq 0}_{k+1,n}under the moment map—translate into sign conditions characterizing the T-dualGrasstopes—images of positroid cells ofGrk,n≥0Gr^{\geq 0}_{k,n}underZ~\tilde {Z}. Moreover, we subdivide the amplituhedron intochambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedronAn,k,2(Z)\mathcal {A}_{n,k,2}(Z)for allZZ. Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization ofAn,k,2\mathcal {A}_{n,k,2}, and Łukowski–Parisi–Spradlin–Volovich’s conjectures onm=2m=2cluster adjacencyand onpositroid tilesforAn,k,2\mathcal {A}_{n,k,2}(images of2k2k-dimensional positroid cells which map injectively intoAn,k,2\mathcal {A}_{n,k,2}). Finally, we introduce new cluster structures in the amplituhedron.

Robust interpolation for dispersed gas‐droplet flows using statistical learning with the fully Lagrangian approach

SummaryA novel methodology is presented for reconstructing the Eulerian number density field of dispersed gas‐droplet flows modelled using the fully Lagrangian approach (FLA). In this work, the nonparametric framework of kernel regression is used to accumulate the FLA number density contributions of individual droplets in accordance with the spatial structure of the dispersed phase. The high variation which is observed in the droplet number density field for unsteady flows is accounted for by using the Eulerian‐Lagrangian transformation tensor, which is central to the FLA, to specify the size and shape of the kernel associated with each droplet. This procedure enables a high level of structural detail to be retained, and it is demonstrated that far fewer droplets have to be tracked in order to reconstruct a faithful Eulerian representation of the dispersed phase. Furthermore, the kernel regression procedure is easily extended to higher dimensions, and inclusion of the droplet radius within the phase space description using the generalised fully Lagrangian approach (gFLA) additionally enables statistics of the droplet size distribution to be determined for polydisperse flows. The developed methodology is applied to a range of one‐dimensional and two‐dimensional steady‐state and transient flows, for both monodisperse and polydisperse droplets, and it is shown that kernel regression performs well across this variety of cases. A comparison is made against conventional direct trajectory methods to determine the saving in computational expense which can be gained, and it is found that times fewer droplet realisations are needed to reconstruct a qualitatively similar representation of the number density field.

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

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 study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.

Several Characterizations of Δh-Doped Special Polynomials Associated with Appell Sequences

The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of Δh tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of Δh Appell polynomials such as Δh Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained.