Theory of Special Polynomials

Theory of Special Polynomials offers a unified exploration of classical and emerging families of special polynomials through algebraic, operational, and computational methods. Beginning with the foundations of orthogonal polynomials, exponential operators, Hermite polynomials, and the monomiality principle, the book develops the theory of Appell and \(\Delta_\lambda\)-Appell sequences, including Bernoulli, Euler, and hypergeometric extensions. It then introduces degenerate versions of hypergeometric Bernoulli and Euler polynomials, establishing their generating functions, recurrence relations, operational identities, matrix representations, and zero distributions. New mixed-type Fibonacci–Mittag-Leffler and Lucas–Mittag-Leffler polynomials are also investigated, with particular attention to their algebraic structure and the asymptotic behavior of their complex zeros. The final chapter connects analytical and numerical techniques with machine learning through polynomial regression and regularization methods. Combining rigorous theory with symbolic computation and graphical analysis, this volume provides a valuable resource for researchers, graduate students, and applied mathematicians interested in special functions, computational mathematics, approximation theory, and data-driven modeling.