In this paper, we consider the degenerate Apostol-type polynomials and compute their zeros numerically and present them graphically. In particular, we explore the application of these polynomials in approximation theory by introducing a family of Szász-type positive linear operators built from degenerate Apostol-type polynomials defined via a generating function. These operators are explicitly given in terms of those polynomials and act on samples of the target function at uniformly spaced nodes. Their basic moment properties show that they preserve constants, reproduce linear behavior up to a bias that is inversely proportional to n, and admit a closed expression for the second moment. Using Korovkin’s theorem, we prove the uniform convergence of these Szaśz-type approximation operators on compact subsets of the nonnegative real line for continuous functions under mild growth conditions. Also, we derive a quantitative error bound in terms of the modulus of continuity. The theory is complemented with numerical examples and computations of absolute error of approximations for the specific choices of parameters, illustrating the approximation behavior and error trends of these operators.

In this work, we explore the defocusing nonlinear Schrödinger equation known as the Camassa–Holm–Nonlinear Schrödinger (CH–NLS) model, which is newly derived in the sense of deformation of hierarchies of integrable systems. This equation captures important features of nonlinear wave propagation in shallow water dynamics, plasma physics, and nonlinear optics. A comprehensive Lie symmetry analysis is carried out to determine the symmetry generators and to derive similarity reductions of the equation. These reductions enable us to construct several exact group-invariant solutions. In addition, we employ Ibragimov’s conservation theorem to derive associated conservation laws based on the identified symmetries. The study also includes an investigation of modulational instability using the framework of linear stability analysis to assess the stability properties of continuous wave solutions. The results provide analytical insights into the interplay between symmetry, conservation, and stability in the CH–NLS model, thereby contributing to the broader understanding of nonlinear dispersive wave phenomena.

This paper presents a systematic study of the mixed degenerate Gould–Hopper type polynomials, beginning with their generating functions and fundamental operational properties. Summation identities, addition formulas, and a determinant representation are developed to reveal the underlying algebraic structure. A computational investigation of the zero distributions is carried out, supported by numerical illustrations and pattern analysis. The results provide insight into the analytical behavior of the family and its structural richness. Concluding remarks discuss theoretical implications and potential directions including asymptotics, orthogonality, and applications in mathematical physics.

Abstract In this paper, we introduce and investigate a new hybrid class of special polynomials, called the multivariate Hermite–Fubini based Appell polynomials of order α . The proposed family is constructed by integrating the combinatorial generating mechanism of Fubini polynomials of order α with the operational structure of three-variable Hermite–Appell polynomials within the framework of the monomiality principle. An explicit generating function for the new polynomial sequence is established, and several fundamental properties are derived. In particular, the operational identities, differential relations, and shift formulas for the multiplicative and derivative operators are derived. Moreover, a determinant representation of the polynomial family using Cramer’s rule is also obtained. Various special cases are also presented to demonstrate how the new polynomials are related to some well-known classical sequences. Theoretical findings are also supported by computational examples and graphical representations to describe the structural properties of the polynomials.

In this paper, we introduce the parametric trigonometric-type U-Fubini-Bernoulli polynomials and the U-Fubini-Bernoulli-type numbers. Then, we explain their properties and their relationships to other families of polynomials, such as Pollaczek polynomials, Laguerre polynomials, and the Hermite polynomials. In the same way, we also examine the Fourier expansions and derive the integral representation of this family of polynomials.

In this work, we offer the novel class of (p, q)-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, (p, q)-derivative properties, (p, q)-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the (p, q)-Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional (p, q)-Hermite-Appell polynomials are provided. Moreover, we acquire a (p, q)-differential operator formula for (p, q)-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of (p, q)-Hermite-Appell, along with two-dimensional (p, q)-Hermite-Appell polynomials.

We develop a Hilfer-adapted Taylor-type framework that is compatible with the natural initial trace of the Hilfer fractional derivative. For 0 < alpha < 1 and beta is an element of [0, 1], we introduce a shifted (alpha,beta)-fractional power series (FPS) with delta = alpha(1-beta) + beta - 1 and define Hilfer-Taylor coefficients via the regularized trace T-n(f) = (I(1-beta)(1-alpha)(D-alpha,D-beta)(n)f)(0+). This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis t(n alpha+delta)/Gamma(n alpha + delta + 1). Using the associated Mittag-Leffler eigenfunction kernel G(alpha,delta)(t, x) = x(delta)E(alpha,delta+1)(t(alpha)x(alpha)), we define fractional Appell-type sequences through a Hilfer-adapted generating identity and establish their main operational properties, including a lowering relation under D-x(alpha,beta) . As an application, we introduce Bernoulli-type objects and derive a convolution recurrence for the corresponding fractional Bernoulli numbers, recovering the classical case when (alpha,beta) = (1, 1).

In this work, a new class of Gould-Hopper Sheffer-A polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer-A sequences. The proposed family was defined through an appropriate exponential generating function involving trigonometric factors and was shown to possess rich algebraic and operational properties. Explicit series representations and determinant forms were derived using Riordan array techniques and Cramer's rule. By employing the monomiality principle, the associated multiplicative and derivative operators were constructed, establishing the quasi-monomial character of the introduced polynomials and leading to the corresponding differential equations. Furthermore, several important subclasses, including Gould-Hopper-Bernoulli-A, Euler-A, Genocchi-A, and Laguerre-A polynomials, were obtained as illustrative examples. These examples demonstrated the unifying nature of the proposed framework and highlighted its potential applicability in operational calculus, special functions, and mathematical physics.

In this article, we define and study a novel class of function sequences based on their relationship with fractional-order difference operators in the Caputo sense. These sequences, known as ∇ _λ ^α -Appell sequences, are constructed for 0< α≤ 1 and λ >0 . The terms of these sequences involve combinations of monomials with fractional powers and exhibit unique properties that extend classical results in the context of fractional calculus and difference operators.

The paper presents a new type of generalized Apostol-type Frobenius-Euler polynomials and numbers with specific order kappa and level m. We establish fundamental identities and properties using generating function techniques, such as summation formulas, differential and integral relations, and addition theorems. Additionally, we explore the connections between these polynomials and the Stirling numbers of the second kind, as well as other polynomial families. Lastly, we derive a differential equation and a recurrence relation for these new classes of polynomials. Finally, we show applications that can be obtained using these polynomials where the graphs of the zero functions and the meshes are displayed.

The main objective of this work is to investigate a novel class of polynomials, called the degenerate-Sheffer polynomials and to explore their various properties. The generating function, explicit representations, quasi-monomiality and certain novel identities involving degenerate Sheffer polynomials are obtained. Also, the degenerate Sheffer polynomials are explored via determinant representation. Further, the degenerate Gould-Hopper-Sheffer polynomials are introduced with the help of degenerate Sheffer and degenerate Gould-Hopper polynomials. Certain fascinating results, such as the generating function, determinant form, multiplicative and derivative operators and many more results for these hybrid forms of the degenerate Sheffer polynomials, are also obtained. Certain examples are considered special cases of degenerate Gould-Hopper-Sheffer polynomials.

In the present paper, we demonstrate 3-variable 2-parameter q-Hermite polynomials via generating functions along with their series definitions, q-derivatives and operational identities, then we deduce some properties for 2-variable 1-parameter q-Hermite polynomials. Also, we present the same mentioned features for multi-index q-Hermite polynomials and their associated formalism. Moreover, we utilize the techniques of quasi-monomial extension to explain and implement q-multiplicative and q-derivative operators for q-Hermite polynomials in three variables and multi-index q-Hermite polynomials. Finally, we present applications that can be derived using these polynomials, where the graphs of the zero functions and the meshes are displayed.

This paper presents a study that builds upon existing research by applying the monomiality principle to generate novel results. The study primarily focuses on the construction and analysis of tangent-based App ell polynomials, exploring their properties in detail, including their explicit and determinant forms, and their compliance with the monomiality principle. The study also investigates specific classes of App ell polynomials namely, the tangent-based Bernoulli, Euler and Genocchi polynomials and derives key outcomes for each. Additionally, the paper provides numerical and graphical representations of these polynomials, facilitating a deeper understanding of their characteristics. This research contributes to the broader field of special polynomials and their applications in various mathematical and scientific contexts.

This article introduces a new class of multi-variate Hermite-Frobenius-Genocchi polynomials and explores various characterizations of these polynomials. We examine their properties, including recurrence relations and shift operators. Using the factorization method, we derive differential, partial differential, and integrodifferential equations satisfied by these polynomials. Furthermore, we present the Volterra integral equation associated with these multi-variate Hermite-Frobenius-Genocchi polynomials, which improves our understanding and application of the factorization method in fields such as physics and engineering.

This study introduces an innovative framework for generalized Hermite-Frobenius-Genocchi polynomials in two variables, parameterized by a single variable. The focus is on providing a comprehensive characterization of these polynomials through various mathematical tools, including generating functions, series expansions, and summation identities that uncover their essential properties. The work extends to the derivation of recurrence relations, the investigation of shift operators, and the formulation of multiple types of differential equations. In particular, the study delves into integro-differential and partial differential equations, employing a factorization technique to develop different forms and solutions. This multifaceted approach not only enhances our understanding of these polynomials, but also lays the groundwork for their further exploration in diverse areas of mathematical research.

The main objective of this work is to investigate a novel class of polynomials, called the Delta h-Sheffer polynomials and to explore their various properties. The generating function, explicit representations, quasi-monomiality, and certain novel identities involving Delta h-Sheffer polynomials are obtained. Also, the Delta h-Sheffer polynomials are explored via determinant representation. Further, the Delta h Gould-Hopper-Sheffer polynomials are introduced with the help of Delta h-Sheffer and Delta h Gould-Hopper polynomials. Certain fascinating results, such as the generating function, determinant form, multiplicative, and derivative operators and many more results for these hybrid form of the Delta h-Sheffer polynomials are also obtained. Certain examples are considered as the special cases of Delta h Gould-Hopper-Sheffer polynomials.

In the domain of specialized mathematical functions, the rising interest in q-calculus continues to attract researchers, offering powerful tools for modeling in fields like quantum computing, non-commutative probability, combinatorics, functional analysis, mathematical physics, and approximation theory. This study introduces the framework of multivariate q-Hermite-based Appell polynomials, employing various q-calculus techniques. Key properties and fresh insights into these polynomials are examined, such as their generating functions, series representations, recurrence relations, q-differential identities, and operational mechanisms. Additionally, it is shown that these polynomials maintain a quasi-monomial structure within the q-calculus context.

In this paper, we define the three parametric types of Apostol-type unified Bernoulli-Euler polynomials. We present fundamental properties of these polynomials through the utilization of their generating functions. Furthermore, we derive the partial derivatives of these polynomials. Subsequently, we introduce bivariate polynomials and determine their zeros, graphical representations, and approximation values for specific parameters.

This paper investigated the fundamental characteristics and uses of a new class of bivariate quantum-Hermite-Appell polynomials. The series representation and generating relation for these polynomials were derived. Also, a determinant representation for these polynomials was derived. Further, important mathematical characteristics were derived, such as $ q $-recurrence relations and $ q $-difference equations. These polynomials' numerical features were methodically examined, providing information on their computational possibilities and the framework of their zeros. A coherent framework was established by extending the study to related families, such as quantum-Hermite Bernoulli, quantum-Hermite Euler, and quantum-Hermite Genocchi polynomials. These discoveries enhance the knowledge of quantum polynomials and their relationships to classical and contemporary special functions.