Key Results on Special Classes of Polynomials

Abstract

In this thesis, we focus on the study and analysis of special classes of polynomials, namely the Bernoulli, Euler, and generalized Laguerre polynomials. We explore both classical and generalized forms of these polynomials, presenting their generating functions, explicit expressions, recurrence relations, and various interconnections through special numbers such as Stirling numbers. A particular emphasis is placed on the generalized versions, including the generalized Bernoulli polynomials, generalized Euler polynomials, and the generalized Laguerre polynomials, highlighting their structure and importance in analytical and combinatorial contexts. This work also discusses their relations with other special functions like the hypergeometric function and the Riemann zeta function, and provides proofs and applications relevant to both theoretical and applied mathematics.