Euler polynomials: An examination from the perspective of umbral calculus and reflections kaniadakis calculus

In this thesis Euler polynomials, which are among the fundamental elements of mathematical structures and have been studied by many mathematicians, are examined along with their relationships to other special families of polynomials that exhibit structural or analytical similarities. These special polynomials, which appear in various fields, hold significant importance in both theoretical research and applied mathematics. Within the scope of this study, the importance of Euler polynomials in classical and modern mathematics is emphasized, and their connections with other types of polynomials are investigated. Furthermore, their applications within Umbral analysis and Kaniadakis analysis are discussed in detail. Accordingly, the main objective of this study is to develop a new generalization of Euler polynomials within the framework of Kaniadakis calculus, starting from their classical form. The Kaniadakis approach describes classical distributions within a broader framework and employs a specific coefficient that determines how far the structure deviates from classical behavior. When this coefficient equals zero, classical results are recovered. Thus, it enables the explanation of structures with different levels of complexity within a unified framework. Within this approach, the Kappa-Euler polynomials, obtained by generalizing Euler polynomials through the \kappa- parameter, present a structure capable of offering new perspectives in both theoretical and applied mathematics.