Special Functions and Orthogonal Expansions with Applications to Nonlinear Dynamics and Quantum Mechanics

这个项目结合了两位研究者在正交展开、特殊函数及其应用等问题上的努力。该方案涉及调查的定性和定量特征正交多项式在几个变量和多项式正交相对于复杂的权函数。利用椭圆函数研究随机过程和摄动理论中正交多项式的谱性质。额外的工作将用于研究降低描述量子力学方程复杂性的算法。特别地，将考虑Pade近似或其径向简化的适用性。相关工作包括复正交多项式的根分布和Pade近似。后者可能有助于理解非线性动力学中的共振现象。虽然这些研究中的一些代表了早期工作的延续，但与多元多项式相关的研究是在许多数学家经过一段时间的深入研究后取得的最新突破。只有在过去的一年中，才获得了经典单变量结果的多元类似物。这些包括多元积分扩展了众所周知的积分和积分。这项研究的复杂性将需要先进的计算支持测试和实验。

This project combines the efforts of two investigators working on problems involving orthogonal expansions, special functions and their applications. The program concerns investigations into qualitative and quantitative features of orthogonal polynomials in several variables and polynomials orthogonal with respect to complex weight functions. Work is also planned on the application of elliptic functions to study spectral properties of orthogonal polynomials arising from stochastic processes and perturbation theory. Additional work will be done investigating algorithms which reduce the complexity of equations describing quantum mechanics. In particular, the applicability of Pade approximants or their radial simplifications will be considered. Related work involves the distribution of roots of complex orthogonal polynomials and Pade approximants. The latter may shed light on the poles of Pade approximants in understanding resonance phenomena in non-linear dynamics. While some of this research represents continuations of earlier work, that relating to multivariate polynomials follows recent breakthroughs following a period of intense research by a number of mathematicians. Only during the past year have multivariate analogues of classical one-variable results been attained. These include multivariate integrals extending the well-known gamma and beta integrals. The complexity of this research will require advanced computational support for testing and experimentation.