Shahid Wani

1 Introduction and Preliminaries Polynomial families form a foundational pillar in applied mathematics, given their multifaceted characterizations—ranging from orthogonality and generating functions to differential expressions, operational techniques, integral representations, and recurrence relations. Due to their versatile nature and diverse applications, their generalizations and extensions continue to garner significant attention across mathematical and physical sciences.

In this article, we introduce a new class of Δ h $$ {\Delta}_h $$ truncated exponential-based Appell polynomials and investigate their key properties and identities. The study further explores their connection with the monomiality principle, providing a deeper algebraic structure. Extensions of this framework lead to the construction of Δ h $$ {\Delta}_h $$ truncated exponential-based Bernoulli, Euler, and Genocchi polynomials, along with their respective properties.

This paper presents a novel family of bivariate two-dimensional ( ) -Hermite–based Appell polynomials, introducing their construction along with explicit examples to illustrate their structure. The study explores key mathematical properties of these polynomials, including explicit forms and determinant-based representations that provide a foundational understanding of their algebraic framework.