A Comprehensive Study of Generating Functions for Laguerre, Legendre, and Gegenbauer Polynomials

Abstract

The study of generating functions for special polynomials provides a unified framework for understanding the structure and applications of polynomial families like Laguerre, Legendre, and Gegenbauer polynomials. These polynomials play essential roles in areas such as quantum mechanics, numerical analysis, and orthogonal function theory. This paper explores the derivations of their generating functions, discusses their properties (such as recurrence relations and orthogonality), and highlights their applications in solving differential equations, series expansions, and computational methods. Through the analysis of generating functions, we gain deeper insights into the analytical and numerical behaviors of these polynomials in various scientific contexts.