Uniform and Exponential Asymptotics with Applications to Orthogonal Polynomials

在这项建议中，将进行若干项目。作为研究的主要领域之一，邓斯特早期发展的涉及联合拐点和极点的一致渐近理论将用于研究非经典正交多项式（重点是Meixner， Pollaczek和Charlier多项式）。这些较早的渐近理论也将用于研究由种群生物学引起的某些奇异摄动边值问题。在此建议中，Dunster还打算将某些经典的一致渐近airy函数展开式（涉及大参数和转折点）求和为收敛表示。新的结果可以直接应用于大阶贝塞尔函数和大阶legendre函数。提出的问题的一个中心和统一的组成部分是明确的和现实的误差界限的发展。更一般地说，这个建议是基于所谓的特殊函数的近似。特殊函数是在物理、工程和数学的许多领域中出现的某些方程的解。这是一个非常经典的数学分支，已经为我们对物理世界的理解做出了很大贡献。关于特殊函数的数值和理论结果在几乎所有的物理科学中仍然非常重要。此外，在过去的十年中，出现了一系列新的和令人兴奋的思想，导致了渐进方法准确性的重大改进，例如数学物理学家迈克尔·贝里（Michael Berry）首创的所谓的“指数渐近”。在这个项目中，邓斯特打算在多个方向上使用和扩展这些想法。该建议的一个方面是获得比以前获得的更广泛的变量范围有效的近似值。这些更强大的近似的一个直接优势是在数值应用中有更大的灵活性，以及具有重要的理论意义。

DMS-9970489In this proposal a number of projects will be undertaken. As oneof the main areas of research, uniform asymptotic theories involving acoalescing turning point and pole, developed earlier by Dunster, will beused in the study of non-classical orthogonal polynomials (with an emphasison the Meixner, Pollaczek and Charlier polynomials). These earlierasymptotic theories will also be used in the study of certain singularlyperturbed boundary value problems arising from population biology. Also inthis proposal Dunster intends to sum certain classical uniform asymptoticAiry function expansions (involving a large parameter and a turning point)into convergent representations. The new results would then have animmediate application to Bessel functions of large order and Legendrefunctions of large degree. A central and unifying component of the proposedproblems is the development of explicit and realistic error bounds.More generally, this proposal is based on the approximation of theso-called special functions. Special functions are solutions of certainequations which appear in many areas of physics, engineering andmathematics. This is a very classical branch of mathematics that hasalready contributed much to our understanding of the physical world.Numerical and theoretical results concerning special functions are still ofgreat importance in virtually all of the physical sciences. Morever, duringthe past decade there have been a series of new and exciting ideas leadingto significant improvements in the accuracy of asymptotic methods, such asthe so-called "exponential asymptotics" pioneered by the mathematicalphysicist Michael Berry. In this project Dunster intends to use and extendthese ideas in a number of directions. One aspect of the proposal is toobtain approximations which are valid for a wider range of variables thanobtained previously. An immediate advantage of these more powerfulapproximations is that there is greater flexibility in their use innumerical applications, as well as having important theoreticalimplications.