Bernoulli polynomials and their reflections on Kaniadakis analysis

Special polynomials are fundamental to many scientific studies and developments, and are widely used in mathematics, physics, engineering, economics and other scientific fields. Among these special polynomials, Bernoulli polynomials form an important class of special polynomials in both classical analysis and number theory. In this study, the fundamental definition, analytical properties and classical results of Bernoulli polynomials will first be reviewed. Subsequently, the relationships with certain special families of polynomials that exhibit structural similarity to these polynomials will be addressed. The relationships in question will be evaluated in terms of generator function structures and specific value calculations. The main objective of this study is to construct a new family of polynomials from both a theoretical and applied mathematical perspective, starting from classical Bernoulli polynomials and generalising them under the Kaniadakis approach. This structure, based on the Kaniadakis approach known as Kappa analysis, is a mathematical framework developed specifically to address the needs arising in fields such as statistical physics, information theory and generalised thermodynamics. In this regard, the aim is to introduce a new structure to the literature through Kappa-Bernoulli polynomials via the κ parameter.