Mathematical Sciences: Orthogonal Polynomials and Some of Their Applications

数学科学：正交多项式及其一些应用

• 批准号: 8714630
• 负责人: Mourad Ismail
• 金额: $ 8.19万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8714630&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Orthogonal; Polynomials; Some

Several specific topics will be addressed in this project. The first of these concerns Wilson polynomials - these are a general class of polynomials governed by four parameters and are related to classical hypergeometric functions (appearing as seems of the initial segments of these functions). Work will be done on asymptotic expansions of polynomials as certain of the parameters increase withound bound. The work is motivated by problems in combinatorial design. A second area of research concerns continued fraction expansions of quotients of hypergeometric functions. This idea can be traced back to work of Gauss and continues to be of interest because convergent positive continued fractions are Stieltjes transforms of the positive measure against which the denominator polynomials are otrhogonal. Ultimately one will want to invert the transform to find the elusive measure. Another direction represents a continuation of the principal investigator's work on Askey-Wilson integrals. Some remarkable applications of special functions have been made to the evaluation of complex definite integrals. The Askey-Wilson example was one in which an unexpected combinatorial evaluation was discovered. The current work, in collaboration with Dennis Stanton, will seek to flesh out the fundamental ideas behind this discovery. They will begin with a particularly simple class of polynomials and seek to connect the integral of products of these polynomials with basic identities found in decompositions of multipartite graphs.

本项目将涉及几个具体的主题。 其中第一个涉及威尔逊多项式-这些是一个 由四个参数控制的一般多项式类， 与经典的超几何函数有关（似乎是 这些功能的初始部分）。 工作会完成的 关于多项式的渐近展开式 参数增大而无约束。 这项工作的动机是 组合设计中的问题。 第二个研究领域是连分数 超几何函数导数的展开式 这个想法 可以追溯到高斯的工作， 因为收敛正连分数是 正测度的Stieltjes变换， 分母多项式是正交的。 最终人们会希望 求逆变换以找到难以捉摸的测度。 另一个方向表示主体的延续 Askey-Wilson积分的研究 一些显著 已将特殊功能应用于 复定积分的计算 询问威尔逊 一个例子是一个意想不到的组合评估， 被发现了 目前的工作，与丹尼斯合作， 斯坦顿将试图充实这背后的基本思想 的发现 它们将开始于一个特别简单的类 多项式，并寻求连接这些产品的积分 多项式的分解中发现的基本单位 多部图