Developing Special Functions tools for contemporary problems in physics

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`Special functions' are mathematical functions that arise from solving `special' problems in physics. It is known that many physical and real-world phenomena modeled using differential equations. Specifically, various quantum systems reduced to the analysis of second-order differential equations with polynomial coefficients. Until a few decades ago, the theory of special functions was considered exhausted with well-established results entirely available to physicists. Recently, however, new trends in physics emerged, for instance: Supersymmetry, Supergravity, Integrable systems, Quantum confined systems, Quantization techniques, Coherent states, and Factorization method (Darboux transformations, Bi-spectrality), that required innovative techniques to study them. Over the past six years, my research interests focused on: 1) Developing exact and approximate solutions to eigenvalues problems in physics. The tools were based on the Asymptotic Iteration Method (AIM); a widely used method developed in collaboration with R. L. Hall and H. Ciftic 2003. 2) Analyzing the analytic solutions of linear differential equations with polynomial coefficients that go beyond the standard classification. This analysis allowed intensive study of different classes of differential equations used extensively in theoretical physics. 3) Analyzing the necessary and sufficient conditions under which higher-order differential equations admits polynomial solutions. This study was particularly useful in enormous applications. 4) Providing a systematic study of Appell hypergeometric series to present concrete ready-to-use formulas to solve problems in physics and engineering. 5) Studying the multivariate orthogonal polynomials, particularly, 2D-Zernike polynomials that are orthogonal 2D polynomials in the unit disc. Consequently, several quantization problems were analyzed and studied. Several new summation formulas and integral representations were introduced and proved in details. The aims and objectives of the present research proposal are: 1) To introduce novel methods based on Special Functions to understand contemporary problems in physics. 2) To exhibit new exact and iterative solutions to physical eigenvalue problems, especially for those where analytic solutions are not possible or not available. 3) To study the different confluent forms of the Heun differential equation and analyze the conditions that permit polynomial solutions. Such studies will lead to a better understanding of the quasi-exactly solvable spectral problems. 4) To extend my early work on Appell series, focusing on recent applications in physics. 5) To elaborate on the iterative aspects of AIM, that yield a highly accurate approximation of eigenvalue problems. 6) To introduce a discrete version of AIM, focusing on analyzing second-order difference equation and discrete orthogonal polynomials.