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.

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.

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by ${\mathcal{W}}_{n,\lambda }^{(\alpha )}(\delta ,\zeta ;\rho ;\mu )$. We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of ${\mathcal{W}}_{n,3}^{(\alpha )}(\delta ,4;3;2)$.

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.

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 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.

In this research, we leverage various $q$-calculus identities to introduce the notion of $q$-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $q$-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $q$-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $q$-calculus.

In this work, a new class of Gould-Hopper Sheffer- $\lambda$ polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer- $\lambda$ 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- $\lambda$, Euler- $\lambda$, Genocchi- $\lambda$, and Laguerre- $\lambda$ 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.