Mathematical Sciences: Orthogonal Polynomials and Some of Their Applications

本项目将讨论几个具体的主题。第一个是威尔逊多项式——这是一类由四个参数控制的多项式，与经典的超几何函数（似乎是这些函数的初始部分）有关。当某些参数在边界内增加时，将对多项式的渐近展开式进行研究。这项工作的动机是组合设计中的问题。第二个研究领域涉及超几何函数商的连分式展开式。这个想法可以追溯到高斯的工作，并且仍然很有趣，因为收敛的正连分式是对正测度的Stieltjes变换，分母多项式是正交的。最终，人们会想要反转变换来找到难以捉摸的度量。另一个方向代表了首席研究员在Askey-Wilson积分上的工作的延续。在复定积分的求值中，给出了一些特殊函数的显著应用。在Askey-Wilson的例子中，发现了一个意想不到的组合计算。目前的工作，与丹尼斯·斯坦顿合作，将寻求充实这一发现背后的基本思想。他们将从一个特别简单的多项式类开始，并试图将这些多项式的积的积分与在多部图分解中发现的基本恒等式联系起来。

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.