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都是用微分方程式建模的。具体地说，各种量子系统归结为分析具有多项式系数的二阶微分方程组。直到几十年前，特殊函数理论还被认为已经穷途末路，物理学家完全可以得到公认的结果。然而，最近物理学出现了新的趋势，例如：超对称性、超重力、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、双谱)，这些都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合作开发的一种广泛使用的方法。*2)分析多项式系数超出标准分类的线性微分方程组的解析解。通过这种分析，可以对理论物理中广泛使用的不同类型的微分方程进行深入的研究。*3)分析高阶微分方程允许多项式解的充要条件。本文的研究具有广泛的应用前景。*4)系统地研究了Appell超几何级数，给出了解决物理和工程问题的具体公式。*5)研究了多元正交多项式，特别是单位圆盘上的2D-Zernike多项式。因此，对几个量化问题进行了分析和研究。引入并详细证明了几个新的求和公式和积分表示。*本研究方案的目的和目标是：*1)引入基于特殊函数的新方法来理解当代物理问题。*2)给出物理本征值问题的新的精确和迭代解，特别是对于那些解析解不可能或不可用的问题。*3)研究Heun微分方程的不同汇合形式，并分析允许多项式解的条件。这样的研究将有助于更好地理解准精确可解谱问题。*4)扩展我早期在Appell级数方面的工作，重点介绍最近在物理学中的应用。*5)详细说明AIM的迭代方面，从而产生对本征值问题的高精度近似。*6)引入AIM的离散版本，重点分析二阶差分方程和离散的正交多项式。**

`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.**