Search Results (78)

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 study is inspired by the rich symmetry and diverse applications of special polynomial families, with a particular focus on the q-Bernoulli polynomials, which have recently emerged as significant tools in bi-univalent function theory. These polynomials are distinguished by their mathematical versatility, analytical manageability, and strong potential for generalization, offering an elegant framework for advancing the study of such functions. In this paper, we introduce a novel subclass of bi-univalent functions defined through q-Bernoulli polynomials. We obtain coefficient estimates for functions in this class and investigate their implications for the Fekete–Szegö functional. Additionally, we present several new results to enrich the theoretical landscape of bi-univalent functions associated with q-Bernoulli polynomials. Full article

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for $|d_2|$ and $|d_3|$, and provide estimates for the Fekete–Szegö functional $|d_3-\xi d_2^2|, \xi \in \mathbb{R}$. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature. Full article

In this paper, based on previous study of type 2 $\omega$-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2 $\omega$-Daehee polynomials. By employing the methods of generating functions and Riordan arrays, we investigate the properties of these higher-order degenerate polynomials in depth and establish identities that relate them to certain special combinatorial sequences. Full article

The central focus of this study is the development and investigation of a generalized subclass of bi-univalent functions, defined using the $q^2$-Srivastava–Attiya operator in conjunction with Bernoulli polynomials. We derive initial coefficient estimates for functions in the newly proposed class and also provide bounds for the Fekete–Szegö functional. In addition to presenting several new findings, we also explore meaningful connections with previously established results in the theory of bi-univalent and subordinate functions, thereby extending and unifying the existing literature in a novel direction. 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

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in $\mathfrak{U}=$$\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$, characterized by Bernoulli polynomials associated with imaginary error functions. For functions belonging to these subclasses, we establish bounds for their initial coefficients. For these classes, we also tackle the Fekete–Szegö problem. Several new results are also obtained as special cases by specifying certain parameter values in the general findings. 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 the era of big data, the security of information encryption systems has garnered extensive attention, particularly in critical domains such as financial transactions and medical data management. While traditional Shamir’s Secret Sharing (SSS) ensures secure integer sharing through threshold cryptography, it exhibits inherent limitations when applied to floating-point domains and high-precision numerical scenarios. To address these issues, this paper proposes an innovative algorithm to optimize SSS via type-specific coding for real numbers. By categorizing real numbers into four types—rational numbers, special irrationals, common irrationals, and general irrationals—our approach achieves lossless transmission for rational numbers, special irrationals, and common irrationals, while enabling low-loss recovery for general irrationals. The scheme leverages a type-coding system to embed data category identifiers in polynomial coefficients, combined with Bernoulli-distributed random bit injection to enhance security. The experimental results validate its effectiveness in balancing precision and security across various real-number types. 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

The grey prediction model is designed to characterize systems comprising both partially known information (referred to as white) and partially unknown dynamics (referred to as black). However, traditional GM(1,1) models are based on linear differential equations, which limits their capacity to capture nonlinear and non-stationary behaviors. To address this issue, this paper develops a generalized grey differential prediction approach based on the Bernoulli equation framework. We incorporate the Bernoulli mechanism with a nonlinear exponent n and a dynamic polynomial-driven term. In this work, we propose a new model designated as BPGM(1,1). Another key innovation of this work is the adoption of a nonlinear least squares direct parameter identification strategy to calculate the exponent and polynomial parameters in the Bernoulli equation, which achieves a higher degree of freedom in parameter selection and effectively circumvents the model distortion issues caused by traditional background value estimation. Furthermore, the Euler discretization method is utilized for numerical solving, reducing the reliance on traditional analytical solutions for linear structures. Numerical experiments indicate that BPGM(1,1) surpasses GM(1,1), NFBM(1,1), and their improved versions. By leveraging the synergistic mechanism between Bernoulli-type nonlinear regulation and polynomial-driven external excitation, this framework significantly enhances prediction accuracy for systems characterized by non-stationary behaviors and multi-scale trends. 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

This paper analyzes vibrations induced by a moving bogie passing through a single-layer railway track model. The emphasis is placed on the possibility of unstable behavior in the subcritical velocity range. All results are presented in dimensionless form to encompass a wide range of possible scenarios. The results are obtained semi-analytically, however, the only numerical step involves solving the roots of polynomial expressions. No numerical integration is used, allowing for the straightforward solution of completely undamped scenarios, as damping is not required for numerical stability. The vibration shapes are presented in the time domain in closed form. It is concluded that increased foundation damping worsens the situation. However, in general, the risk of instability in the subcritical velocity range for a moving bogie is lower than that of two moving masses, particularly for higher mass moments of inertia of the bogie bar and primary suspension damping. The study also examines how the results change when a Timoshenko-Rayleigh beam is considered instead of an Euler-Bernoulli beam. Although some cases may appear academic, it is demonstrated that instability in the supercritical velocity range cannot be assumed to be guaranteed. Full article