(untitled) (Leadership, Education, Personality: An Interdisciplinary Journal, ISSN: 2524-6178, 2025)
This study explores the formulation and properties of generating functions (producing functions) for specific families of polynomials, focusing on their derivation, mathematical properties, and applications. Generating functions serve as powerful tools in both theoretical and applied mathematics, providing a systematic approach to encapsulate polynomial sequences. The study examines the generating functions for classical polynomials such as the Legendre, Chebyshev, Hermite, and Laguerre polynomials, as well as their generalized counterparts. We delve into methods of derivation, including series expansions, integral representations, and differential equations. The properties of these generating functions, such as recurrence relations, orthogonality, and asymptotic behavior, are analyzed in detail. Additionally, we explore the practical applications of generating functions in solving combinatorial problems, differential equations, and in mathematical physics, particularly in areas such as quantum mechanics and fluid dynamics. The study also discusses computational methods for evaluating these generating functions, highlighting their relevance in numerical analysis and algorithm development. By providing a comprehensive overview of producing functions for specific polynomials, this work aims to facilitate a deeper understanding of their role in both pure and applied mathematical research.