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

这个项目结合了两个研究者的努力 研究涉及正交展开的问题，特别是 功能及其应用。 该方案关注 调查的定性和定量特征， 多元正交多项式与多项式 关于复权重函数正交。 工作是 还计划对椭圆函数的应用进行研究 正交多项式的谱性质 随机过程和扰动理论。 另外的工作将做调查算法 这降低了描述量子的方程的复杂性， 力学 特别是，Pade的适用性 将考虑近似或其径向简化。 相关的工作涉及复形根的分布 正交多项式和Pade逼近。 后者可能 揭示了帕德逼近的极点， 非线性动力学中的共振现象。 虽然其中一些研究代表了 早期的工作，涉及多元多项式如下 经过一段时间的紧张研究， 一些数学家。 仅在过去一年中， 经典单变量结果的多变量类似物被 达到。 这些包括多元积分， 著名的伽玛和贝塔积分 这项研究的复杂性将需要先进的 为测试和实验提供计算支持。

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.