Search Results (164)

This study offers a comprehensive generalization of the Gould–Hopper polynomials and their Appell-type analogs. Employing the quasi-monomiality approach, we delineate fundamental analytical characteristics, including recurrence relations, associated multiplicative and differential operators, and governing differential equations. Additionally, we derive series representations and determinantal expressions for this newly defined polynomial family. Within this framework, several significant subclasses are introduced and examined, such as the generalized Gould–Hopper-based Appell polynomials. The formulation is further extended using fractional operator techniques to explore their intrinsic structural attributes. Moreover, we construct and investigate new families, namely, the generalized Gould–Hopper-based Bernoulli, Gould–Hopper-based Euler, and Gould–Hopper-based Genocchi polynomials, emphasizing their operational and algebraic properties. Collectively, these findings advance the theory of special functions and provide a foundation for potential applications in mathematical physics and the study of differential equations. Full article

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential operators. By confirming the quasi-monomial structure, we further deduce the governing differential equation for the proposed family. In addition, we present both a series expansion and a determinant formulation, providing complementary representations that are useful for symbolic manipulation and computation. As special cases, we introduce and study subfamilies arising from this setting, namely, extended truncated exponential versions of the Bernoulli, Euler, and Genocchi polynomials, and discuss their structural identities and operational behavior. Overall, these developments broaden the theory of special polynomials and furnish tools relevant to problems in mathematical physics and differential equations. Full article

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

The free vibration of stiffened plates analyzed using classical plate–beam theoretical theory (PBM) simplified the vibrations of stiffeners parallel to the plane of the stiffened plate as the first-order torsional vibration of the stiffener cross-section. This simplification introduces errors in both the natural frequencies and mode shapes of the structure for stiffened plates with relatively tall stiffeners. To mitigate the issue previously described, this paper proposes an enhanced plate–beam theoretical model (EPBM). The EBPM decouples stiffener deformation into two components: (1) bending deformation along the transverse direction of the stiffened plate, governed by Euler–Bernoulli beam theory, and (2) transverse deformation of the stiffeners, modeled using thin plate theory. Virtual torsional springs are introduced at the stiffener–plate and stiffener–stiffener interfaces via penalty function method to enforce rotational continuity. These constraints are transformed into energy functionals and integrated into the system’s total energy. Displacement trial functions constructed from Chebyshev polynomials of the first kind are solved using the Ritz method. Numerical validation demonstrates that the EBPM significantly improves accuracy over the BPM: errors in free-vibration frequency decrease from 2.42% to 0.63% for the first mode and from 9.79% to 1.34% for the second mode. For constrained vibration, the second-mode error is reduced from 4.22% to 0.03%. This approach provides an effective theoretical framework for the vibration analysis of structures with high stiffeners. Full article

Soft pneumatic actuators (SPAs) are gaining attention in the field of soft robotics due to their lightweight, highly flexible, and safer interaction while operated under an unstructured environment. They are easy to fabricate, produce high output force, and are relatively very inexpensive compared to other soft actuators. However, accurate prediction of their nonlinear bending behavior is one of the main challenges, which is mainly due to the complex material properties and high deformation patterns. Therefore, this study focused on a hybrid approach that accurately captures the bending behavior of a single-chambered SPAs. This approach integrates physics-based modeling (finite element analysis (FEA) and analytical modeling) with a data-driven (polynomial regression modeling) approach to analyze the bending of single-chambered SPAs. Initially, four different hyperelastic material models (Neo-Hookean, Yeoh, Arruda–Boyce, and Ogden) were tested using FEA to analyze how material selection affects the SPA response. It is found that the Arruda–Boyce model generates the highest bending of 101° at 30 kPa pressure, while the other models consistently underestimated deformation at higher pressures. Further, an enhanced mathematical or analytical model was developed using Euler and Timoshenko beam theory with certain assumptions, such as neutral axis shifting, chamber ballooning, and shear deformation. These assumptions significantly improve the prediction accuracy and generate a bending angle of 99°at 30 kPa, which closely matches FEA bending. Further, a polynomial regression-based machine learning (ML) model was trained using analytical or mathematical bending data for faster output prediction. This data-driven approach achieves very high accuracy in the validation range, with an average absolute percentage deviation of only 0.002%. Additionally, comparison with the analytical results showed a mean absolute error (MAE) of 0.00180°, root mean squared error (RMSE) of 0.00205°, and coefficient of determination (R²) value of 0.999999808. Overall, integrating physics-based modeling with a data-driven approach provides a reliable and scalable method for SPA design. It provides practical information on material selection, analytical correction, and ML modeling, which will reduce the need for time-consuming prototyping. Finally, this hybrid approach can help to accelerate the development of soft robotic grippers, rehabilitation tools, and other bio-inspired actuation systems. Full article

Using Hoppe’s formula, we derive two identities that relate the powers and derivatives of the generating function for Fubini polynomials. As applications, we obtain several identities involving Fubini polynomials, including identities for Sums of Products of Fubini polynomials. These results refine and extend those established in two previous papers by other authors. Furthermore, we prove conjectures posed in one of these papers and derive several congruence identities for Fubini numbers. Full article

The Kronecker product has been commonly seen in various scientific fields to formulate higher-dimensional spaces from lower-dimensional ones. This paper presents a generalization of the Cannon–Kronecker product bases by introducing generalized Kronecker product bases of polynomials within an analytic framework. It investigates the convergence behavior of infinite series formed by these generalized products in various polycylindrical domains, including both open and closed configurations. The paper also delves into essential analytic properties such as order, type, and the ${\mathbb{T}}_{\rho }$-property to analyze the growth and structural characteristics of these bases. Moreover, the theoretical insights are applied to a range of classical special functions, notably Bernoulli, Euler, Gontcharoff, Bessel, and Chebyshev polynomials. 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

This paper explores the eigenfunctions of specific Laguerre-type parametric operators to develop multi-parametric models, which are associated with a class of the generalized Mittag-Leffler type functions, for dynamical systems and population dynamics. By leveraging these multi-parametric approaches, we introduce new concepts in number theory, specifically those involving multi-parametric Bernoulli and Euler numbers, along with other related polynomials. Several numerical examples, which are generated by using the computer algebra program Mathematica© (Version 14.3), demonstrate the effectiveness of the models that we have presented and analyzed in this paper. Full article

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them as $\kappa$-polynomials. Through the defined concepts of Kaniadakis calculus, we derive explicit formulas, summation formulas, and addition formulas for the polynomials discussed in the present paper. We also present the Volkenborn integral and the fermionic p-adic integral representations in terms of the $\kappa$-Stirling polynomials of the second kind, bivariate $\kappa$-Bell polynomials, $\kappa$-Bernoulli polynomials of higher order, and $\kappa$-Euler polynomials of higher order. We establish some formulae, including old and new polynomials. Finally, we investigate determinantal representations for the $\kappa$-Euler polynomials and the $\kappa$-Bernoulli polynomials. Full article

This study demonstrates the effectiveness of the differential transform method (DTM) in solving complex solid mechanics problems, focusing on static analysis of beams under various loads and boundary conditions. For cantilever beams (BSM1), DTM provided exact polynomial solutions for deflections and slopes: a cubic solution for concentrated end loads, a quadratic distribution for applied moments, and a fourth-degree polynomial for uniformly distributed loads, all matching established theoretical results. For simply supported beams (BSM2), DTM yielded solutions across two intervals for midspan concentrated forces, though required corrective terms for applied moments due to discontinuities. Under uniform loading, the method produced precise polynomial solutions with maximum deflection at midspan. Key advantages include DTM’s high-precision analytical solutions without additional approximations and its adaptability to diverse loading scenarios. However, for cases with pronounced discontinuities like concentrated moments, supplementary methods (e.g., Green’s functions) may be needed. The study highlights DTM’s potential for extension to nonlinear or dynamic problems, while software integration could broaden its engineering applications. This study demonstrates, for the first time, how DTM yields exact polynomial solutions for Euler–Bernoulli beams under discontinuous loads (e.g., concentrated moments), overcoming limitations of traditional numerical methods. The method’s analytical precision and avoidance of discretization errors are highlighted. Traditional methods like FEM require mesh refinement near discontinuities (e.g., concentrated moments), leading to computational inefficiencies. DTM overcomes this by providing exact polynomial solutions with corrective terms, achieving errors below 0.5% with only 4–5 series terms. Full article

We study a class of continued fraction transformations where the partial numerators are quadratic polynomials and the denominators are linear or constant. Using the Bauer–Muir transform, we establish two theorems that yield structurally distinct but equivalent continued fractions—one with rational coefficients and another with alternating forms. These transformations provide a unified framework for evaluating and simplifying continued fractions, including classical identities such as one of Euler, a recent result by Campbell and Chen, and several conjectures from the Ramanujan Machine involving $\pi$ and $\log 2.$ We conclude by discussing the potential extension of our methods to more general polynomial cases. 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 paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article

This paper introduces fully Dowling polynomials of the first and second kinds, which are degenerate versions of the ordinary Dowling polynomials. Then, several important identities for these degenerate polynomials are derived. The relationship between fully degenerate Dowling polynomials and fully degenerate Bell polynomials, degenerate Bernoulli polynomials, degenerate Euler polynomials, and so on is obtained using umbral calculus. Full article