Developing Special Functions tools for contemporary problems in physics

“特殊函数”是在解决物理学中的“特殊”问题时产生的数学函数。众所周知，许多物理和现实世界的现象使用微分方程建模。具体来说，各种量子系统简化为具有多项式系数的二阶微分方程的分析。 直到几十年前，特殊函数的理论被认为已经耗尽，物理学家完全可以得到公认的结果。然而，最近出现了物理学的新趋势，例如：超对称性，超引力，可积系统，量子限制系统，量子化技术，相干态和因子化方法（达布变换，双谱），这需要创新的技术来研究它们。 在过去的六年里，我的研究兴趣集中在： 1)发展物理学中本征值问题的精确和近似解。这些工具基于渐近迭代法（AIM），这是与R。L. Hall和H. Ciftic 2003. 2)分析超出标准分类的多项式系数线性微分方程的解析解。这种分析允许深入研究在理论物理学中广泛使用的不同类别的微分方程。 3)分析了高阶微分方程存在多项式解的充要条件。这项研究在大量应用中特别有用。 4)提供Appell超几何级数的系统研究，以提出具体的现成公式来解决物理和工程问题。 5)研究了多元正交多项式，特别是二维Zernike多项式，它是二维单位圆上的正交多项式。因此，对几个量化问题进行了分析和研究。引入并证明了几个新的求和公式和积分表示。 本研究提案的目的和目标是： 1)介绍基于特殊函数的新方法，以理解物理学中的当代问题。 2)展示物理本征值问题的新的精确解和迭代解，特别是对于那些解析解不可能或不可用的问题。 3)研究Heun微分方程的不同合流形式，分析其存在多项式解的条件。这些研究将有助于更好地理解拟精确可解谱问题。 4)为了扩展我早期对Appell系列的研究，重点是最近在物理学中的应用。 5)详细说明AIM的迭代方面，它产生了本征值问题的高度精确的近似。 6)介绍了离散格式的AIM，重点分析了二阶差分方程和离散正交多项式。

`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.