Search Results (84)

In this article, we present new theoretical findings on specific polynomials that generalize the concept of telephone numbers, namely, Telephone polynomials (TelPs). Several new formulas are developed, including expressions for higher-order derivatives, repeated integrals, and moment formulas of TelPs. Moreover, we derive explicit connections between the derivatives of TelPs and the two classes of symmetric and non-symmetric polynomials, producing many formulas between these polynomials and several celebrated polynomials such as Hermite, Laguerre, Jacobi, Fibonacci, Lucas, Bernoulli, and Euler polynomials. The inverse formulas are also obtained, expressing the derivatives of well-known polynomial families in terms of TelPs. Furthermore, some novel linearization formulas (LFs) with some classes of polynomials are established. Finally, some new definite and indefinite integrals of TelPs are established using some of the developed relations. Full article

This paper introduces a novel multidimensional half-discrete Hardy–Hilbert-type inequality that simultaneously addresses several key extensions in the literature. The inequality incorporates a general parameterized kernel involving a scalar term and the $\beta$-norm of a vector, and replaces the traditional discrete coefficient with a partial sum. Under suitable parameter conditions, the resulting inequality is sharper and preserves the optimal constant factor. The proof employs a systematic combination of weight-function techniques, parameter introduction, real-analysis methods, and the Euler–Maclaurin summation formula. Equivalent characterizations of the best possible constant are provided, and several meaningful corollaries are deduced, thereby unifying and generalizing a series of earlier inequalities. Full article

It is well known that the Cahn–Hilliard equation satisfies the energy dissipation law and the mass conservation property. Recently, the radial basis function–finite difference (RBF–FD) approach and its numerous variants have garnered significant attention for the numerical solution of surface-related problems, owing to their intrinsic advantage in handling complex geometries. However, existing RBF–FD schemes generally fail to preserve mass conservation when solving the Cahn–Hilliard equation on smooth closed surfaces. In this paper, based on an $L^2$ projection method, two numerically efficient RBF–FD schemes are proposed to achieve mass-conservative numerical solutions, which are demonstrated to preserve the mass conservation law under relatively mild time-step constraints. Spatial discretization is performed using the RBF–FD method, while based on the convex splitting method and a linear stabilization technique, the first-order backward Euler formula (BDF1) and the second-order Crank–Nicolson (CN) scheme are employed for temporal integration. Extensive numerical experiments not only validate the performance of the proposed numerical schemes but also demonstrate their ability to utilize mild time steps for long-term phase-separation simulations. Full article

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also provide formulas for the higher-order derivatives of Cauchy polynomials and obtain corresponding formulas and identities for poly-Cauchy polynomials. Furthermore, we introduce a multiparameter framework for poly-Cauchy polynomials, unifying earlier generalizations like shifted poly-Cauchy numbers and polynomials with a q parameter. Full article

Just as Gauss’s interpretation of complex numbers as points in a number plane in the form of a suitably formulated axiom found its way into the vector representation of Fourier transforms, this is the case with Euler’s formula and Riemann’s Zeta function considered here. The description of the connection between variables through complex numbers as it is given in Euler’s formula and emphasized by Riemann is reflected here with great flexibility in the introduction of non-classically generalized complex numbers and the vector representation of the generalized Zeta function based on them. For describing such dependencies of two variables with the help of generalized complex numbers, we introduce manifolds underlying certain Lie groups as level sets of norms, antinorms or semi-antinorms. No undefined or “imaginary” quantities are used for this. In contrast to the approach of Hamilton and his numerous successors, the vector-valued vector product of non-classically generalized complex numbers is commutative, and the whole number system satisfies a weak distributivity property as considered by Hankel, but not the strong one. Full article

This work breaks a 180-year-old framework created by Hamilton both with regard to the use of imaginary quantities and the definition of a quaternion product. The general quaternionic algebraic structure we are considering was provided by the author in a previous work with a commutative product and will be provided here with a non-commutative product. We replace the imaginary units usually used in the theory of quaternions by linearly independent vectors and the usual Hamilton product rule by a Hamiltonian-adapted vector-valued vector product and prove both a new geometric property of this product and a vectorial adopted Euler type formula. Full article

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given. Full article

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

This study introduces a novel generalized class of special polynomials using a fractional operator approach. These polynomials are referred to as the generalized Gould–Hopper–Bell-based Appell polynomials. In view of the operational method, we first introduce the operational representation of the Gould–Hopper–Bell-based Appell polynomials; then, using a fractional operator, we establish a new generalized form of these polynomials. The associated generating function, series representations, and summation formulas are also obtained. Additionally, certain operational identities, as well as determinant representation, are derived. The investigation further explores specific members of this generalized family, including the generalized Gould–Hopper–Bell-based Bernoulli polynomials, the generalized Gould–Hopper–Bell-based Euler polynomials, and the generalized Gould–Hopper–Bell-based Genocchi polynomials, revealing analogous results for each. Finally, the study employs Mathematica to present computational outcomes, zero distributions, and graphical representations associated with the special member, generalized Gould–Hopper–Bell-based Bernoulli polynomials. Full article

A generalized buckling solution for anisotropic laminated composite fixed–free columns under axial compression is developed using the critical stability matrix. The axial, coupling, and flexural equivalent stiffness coefficients of the anisotropic layup are determined from the generalized constitutive relationship through the static condensation of the composite stiffness matrix. The derived formula reduces down to the Euler buckling equation for isotropic and some special laminated composites. The analytical results are verified against finite element solutions for a wide range of anisotropic laminated layups yielding high accuracy. A parametric study is conducted to examine the effects of ply orientations, element thickness, finite element type, column size, and material properties. Comparisons with numerical results reveal a very high accuracy across the entire parametric profile and a linear correlation between the percentage error and a non-dimensional condensed parameter is extracted and plotted. Full article

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs. Full article

Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta-functions by considering some generalizations of the Eisenstein series of level $2\varkappa$ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of ${\sigma }_s\left(n\right)$, the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas. Full article

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class $C^{2k}\left([a,b]\right)$. A stronger version of the estimation of the rest for functions of class $C^n\left([a,b]\right)$ is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure). Full article

In this paper, we present a new integral representation for the Jacobi polynomials that follows from Koornwinder’s representation by introducing a suitable new form of Euler’s formula. From this representation, we obtain a fractional integral formula that expresses the Jacobi polynomials in terms of Gegenbauer polynomials, indicating a general procedure to extend Askey’s scheme of classical polynomials by one step. We can also formulate suitably normalized Fourier–Jacobi spectral coefficients of a function in terms of the Fourier cosine coefficients of a proper Abel-type transform involving a fractional integral of the function itself. This new means of representing the spectral coefficients can be beneficial for the numerical analysis of fractional differential and variational problems. Moreover, the symmetry properties made explicit by this representation lead us to identify the classes of Jacobi polynomials that naturally admit the extension of the definition to negative values of the index. Examples of the application of this representation, aiming to prove the properties of the Fourier–Jacobi spectral coefficients, are finally given. Full article

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems. Full article