Developing Special Functions tools for contemporary problems in physics

“特殊函数”是解决物理中“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都是用微分方程来建模的。具体来说，各种量子系统被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，人们还认为特殊函数理论已经穷尽了，物理学家完全可以得到完善的结果。然而，最近出现了一些新的物理学趋势，例如：超对称、超引力、可积系统、量子受限系统、量子化技术、相干态和因式分解方法（达布变换、双光谱），这些都需要创新的技术来研究。***在过去的六年里，我的研究兴趣主要集中在：***1)发展物理特征值问题的精确和近似解。工具基于渐近迭代法（AIM）；与r.l. Hall和h.c iftic 2003年合作开发的一种广泛使用的方法。***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。这种分析使人们能够深入研究理论物理中广泛使用的不同类型的微分方程。***3)分析高阶微分方程存在多项式解的充分必要条件。这项研究在大量应用中特别有用。***4)对阿佩尔超几何级数进行了系统的研究，为解决物理和工程问题提供了具体的实用公式。***5)研究多元正交多项式，特别是单位圆盘上的二维正交多项式2D- zernike多项式。因此，对几个量化问题进行了分析和研究。介绍并详细证明了几个新的求和公式和积分表示。***本研究计划的目的和目标是：***1)引入基于特殊函数的新方法来理解当代物理问题。***2)展示物理特征值问题的精确和迭代的新解，特别是对于那些不可能或不可用解析解的问题。***3)研究Heun微分方程的不同合流形式，并分析多项式解存在的条件。这样的研究将有助于更好地理解准精确可解的光谱问题。***4)扩展我早期关于Appell系列的工作，专注于最近在物理学中的应用。***5)详细说明AIM的迭代方面，它产生了特征值问题的高度精确的近似。***6)引入离散版本的AIM，重点分析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.**