Abstract

Special functions play an important role in mathematics, physics, engineering, and applied sciences. Functions such as Gamma, Beta, Bessel, Legendre, Hypergeometric, and Hermite functions arise naturally in the solutions of differential equations, integral equations, and boundary value problems. The present study focuses on selected topics in the theory of special functions, including their properties, generating functions, recurrence relations, orthogonality, and applications in scientific problems. The study also examines how these functions are applied in heat conduction, wave propagation, quantum mechanics, and mathematical modeling. Analytical methods are used to derive identities and relationships among special functions. The research highlights the significance of special functions in solving practical and theoretical problems efficiently.