This book presents a collection of contemporary research focused on special functions, Fourier series expansions, generalized polynomials, and functional operators, emphasizing their growing role in modern mathematical analysis and applications. The chapters cover a wide range of specialized topics, including extensions of Appell-type polynomials via fractional operators, Fourier developments of periodic Bernoulli and Euler functions in connection with the Riemann zeta function, and new families of polynomials within the framework of golden calculus. Further contributions explore central Bell-based type Euler polynomials, central Fubini numbers and polynomials, and generalized Bernoulli-Fibonacci polynomials of arbitrary order. The volume also provides a rigorous study of the Hardy–Littlewood operator on the real line, particularly within weighted Sobolev spaces, highlighting the interaction between operator theory and the algebraic structures of special polynomials. This work offers both theoretical insights and applied tools, making it a valuable resource for researchers in mathematical analysis, special functions, operator theory, and fractional calculus.