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In this paper, we study the general rogue wave solutions and their patterns in the vector (or M -component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev–Petviashvili reduction method, we derive an explicit solution for the rogue wave expressed by τ functions that are determinants of K × K block matrices ( 1 ≤ K ≤ M ) with an index jump of M + 1 . Patterns of the rogue waves for M = 3 , 4 and K = 1 are thoroughly investigated. It is found that when one of the internal parameters is large enough, the wave pattern is linked to the root structure of a generalized Wronskian–Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system, and many others. Moreover, the generalized Wronskian–Hermite polynomial hierarchy includes the Yablonskii–Vorob’ev polynomial and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang et al. for the scalar NLS equation and the Manakov system. It is noted that the case M = 3 displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of M = 3 , 4 . An excellent agreement is achieved.

In this paper, we employ an umbral method to reformulate the 3-variable Hermite polynomials and introduce the 4-parameter 3-variable Hermite polynomials. We also obtain some new properties for these polynomials. Moreover, some special cases are discussed and some concluding remarks are also given.

Several adaptive filters have recently incorporated the Maximum Versoria criteria (MVC) and Blake Zisserman techniques to demonstrate their resilience to impulsive noise and non-Gaussian interference. In scenarios involving nonlinear system identification expressing sparse characteristics, the performance of these algorithms degrade when dealing with colored input signals. This manuscript presents the design of a nonlinear adaptive algorithm in the presence of impulsive noise by integrating a Hermite function polynomial in the functional link network, incorporating the Blake Zisserman function as a robust function. Additionally, this script introduces a zero-attracting affine projection Blake Zisserman-based Hermite functional link network (ZAB-HFLN) to model a nonlinear sparse system with impulsive or non-Gaussian noise disturbances, associated with the input as a colored signal. The l 1 regularization term is utilized in the algorithm to effectively model the system along with a sparsity parameter in this approach. Furthermore, a re-weighted ZAB-HFLN (RZAB-HFLN) is incorporated in this study, which integrates a log sum regularization parameter into the cost term function. This addition addresses the challenge of withstanding changing sparsity levels in the desired nonlinear system. The experimental outcomes clearly demonstrate the effectiveness and performance of the proposed algorithm in representing nonlinear systems, particularly when considering the input as colored signals. In addition, the nonlinear acoustic feedback paths of a behind-the-ear (BTE) hearing aid are also modelled employing the proposed techniques.

A new model to accurately obtain the contact stiffness of the mechanical joint surface is established, which satisfies the continuous smooth contact characteristics of the asperity and also considers the influence of the asperity interaction. In order to satisfy the continuous and smooth contact characteristics of the asperity, the interpolation interval of the Hermite polynomial function is improved to make the asperity contact stiffness curve smoothly transition in the elastic, elastoplastic and completely plastic stages. Further, the contact situation of the joint surface considering the asperity interaction is explored through micro-contact analysis and mechanism research. Finally, using statistical principles, the microscopic contact model is extended to the entire joint surface, and the contact stiffness model of the joint surface is established. The simulation results show that the asperity stiffness model satisfies the continuous smooth contact characteristics. Surface roughness is the main factor that affects the contact stiffness, and the influence of the asperity interaction cannot be ignored.

A noteworthy advancement within the discipline of q-special function analysis involves the extension of the concept of the monomiality principle to q-special polynomials. This extension helps analyze the quasi-monomiality of many q-special polynomials. This extension is a helpful tool for considering the quasi-monomiality of several q-special polynomials. This study aims to identify and establish the characteristics of the 2-variable q-Hermite–Appell polynomials via an extension of the concept of monomiality. Also, we present some applications that are taken into account.

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials.

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.

Model of the extended Dunkl oscillator based on Milovanović generalized Hermite polynomials on radial rays is discussed. Simple explicit realization of creation and annihilation operators in terms of difference-differential operators, coherent states are investigated.

Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and λ-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach.